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Order of Operations

GETTING STARTED

Order of Operations

BEDMAS stands for (Brackets, Exponents, Division, Multiplication, Addition and Subtraction). This is an easy way to know in which order to perform arithmetic operations. Must follow the order as they occur from left to right.

     Example:-

                     i) 5 x 3 + 6 = 15 + 6 = 21       

                      Multiply the first two terms and then do the addition.

                    ii) 5 x (3 + 6) = 5 x 9 = 45       

                     Solve terms inside the brackets and then multiply.

                   iii) 4 x 3³ = 4 x 27 = 108

                   iv) 4 ÷ 2 x 6 = 2 x 6 = 12            

                      Division first and then multiplication.

                    v) 6 ÷ (4 x 5) = 6/20 = 3/10      

                      Solve terms inside the brackets and then divide.

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Simplify the Following

Show all work

  1. 5² - 1 = ___                                          [Final Answer : 24 ]

  2. 27 ÷ 3 + 4 = ___                                 [Final Answer : 13 ]

  3. 6 ÷ (4 ÷ 5) = ___                                 [Final Answer : 15/2 ]

  4. 18 ÷ 9 + 4 = ___                                 [Final Answer : 6 ]

  5. 5 + 7 – 6 ÷ 2 +11 = ___                     [Final Answer : 20 ]

  6. 6² ÷ (5 + 2²) – 1 = ___                       [Final Answer : 3 ]

  7. (8² – 4 x 3) / (2⁴ – 3) = ___              [Final Answer : 4 ]

  8. 5 x 3³ - 6(11 + 2²) = ___                    [Final Answer : 45 ]

  9. 55 - 3[(4 + 3) - (18 ÷ 3)]  = ___        [Final Answer : 52 ]

  10. 3(7 - 3)³ + 130 ÷ (14 - 1) = ___        [Final Answer : 202 ]

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Addition & Subtraction of Integers

Addition and Subtractions of Integers
Addition of Integers
Method 1: Using the Number Line

     Example:-

                     i) - 4 + 9 = +5     

                     From  -4 on the number line count 9 units to the right.

                    ii) 2 + (- 7) = -5  

                     From  +2 on the number line count 7 units to the left.

Method 2: Absolute Value

Case 1- Same Sign: Use the common sign when adding the absolute values.

     Example:-

                   i) - 4 + (- 6) = - 10     

                   The common sign is negative as both numbers are negative.

                   ii) 315+ 4 = 319       

                    The common sign is positive as both numbers are positive.

                   iii) -5 + (-8.5) = -13.5 

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Case 2 - Different Signs: Subtract the smaller from the larger absolute number. Sign with the larger absolute value would be used. Adding values of equal magnitude of opposite signs is 0.

     Example:-

                   i) 6 + (- 8) = - 2    

                   Smaller absolute number, 6 is being subtracted from the larger absolute number, 8. The sign of the larger absolute number is used (In our case 8).  

                   ii) -5 + 6 = 1

                   iii) -5 + 5 = 0

                    As stated above, adding numbers of equal magnitude of opposite sign is 0. This can be seen in the above example (iii).

Subtraction of Integers

To subtract integers, add the opposite. The examples below show this procedure.

     Example:-

                     i) 24 - 19 = 24 + (- 19) = + 5

                    ii) 4 - (- 5) = 4 + 5 = + 9

                   iii) 20 + (- 7) - 10 = 13 + (- 10) = + 3

Solve

Show all your Work

  1. 8 + (- 5) = ___                              [Final Answer : 3 ]

  2. - 4 + (- 6) = ___                            [Final Answer : -10 ]

  3. (- 7) + (- 3) = ___                         [Final Answer : -10 ]

  4. 3 + (- 8) = ___                              [Final Answer :  -5 ]

  5. - 2 + 5 + (- 8) = ___                     [Final Answer : -5 ]

  6. 8 + (- 9) + (- 11) + 11 = ___       [Final Answer : -1 ]

  7. 12 - 9 = ___                                  [Final Answer: 3 ]

  8. - 6 - 9 = ___                                  [Final Answer: - 15 ]

  9. 23 - (- 7) = ___                             [Final Answer: 30 ]

  10. 8 + (4 - 9) - (- 3 - 8) = ___           [Final Answer: 14 ]

  11. - 56 + (- 4) - (- 60) = ___             [Final Answer:  0 ]

Multiplication & division of Integers

Multiplication & Division of Integers

In this section one must pay very close attention to the signs when performing the multiplication and division operations. There are rules pertaining to this which need to be followed in order to determine the correct sign of the final answer.

Multiplication of Integers

Rule 1 - The product/multiplication of a negative number by a positive number is always negative.

Rule 2 - The product/ multiplication of two negative/two positive numbers (even number of negative terms) is always positive.

     

     Example:-

                     i) (+ 3)(+ 5) = + 15

                     Since both are positive numbers, the product is positive 15.

                    ii) (- 3)(10) = - 30

                    Since there is one negative and one positive number, the product is - 30.

                   iii) (- 2)(- 3)(- 4) = - 24

                    As seen the above contains an odd number of negatives, thus the product is - 24.

                   iv) 12(- 1)(- 3) = + 36

                     As it contains an even number of negative terms being multiplied by another positive term, thus results in a product of + 36.

Division of Integers

Rule 1 - The quotient of a negative number by a positive number (or vice versa) is always negative.

Rule 2 - The quotient of two positive / two negative numbers is always positive.  

Multiplying the denominator by the quotient results in the numerator. This is a really good way of checking your results.

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     Example:-

                     i) 25 ÷ (- 5) = - 5

                     As seen, rule 1 applies resulting in - 5. This can be checked by multiplying the quotient by the denominator which in this case clearly equals the numerator + 25.

                    ii) - 56 ÷ + 2 = - 28

                      As seen, rule 1 clearly applies , resulting in - 28. 

                   iii) - 44 ÷ (- 11) = + 4

                     As seen, rule 2 applies resulting in + 4. This can be checked by multiplying the quotient by the denominator which in this case clearly equals the numerator - 44.

                   iv) 35 ÷ 7 = + 5 

                     As seen, rule 2 clearly applies , resulting in + 5. 

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Solve

Show all your Work

  1. (- 4)(- 2) = ___                             [Final Answer : +8 ]

  2. 3 x 7 = ___                                   [Final Answer : +21 ]

  3. (- 8)(- 7) = ___                             [Final Answer : +56 ]

  4. (-15)(- 3)(-1) = ___                     [Final Answer : - 45 ]

  5. 5(- 3)(- 2) - (- 5)(2) = ___           [Final Answer : + 40 ]

  6. 25 ÷ (- 5) = ___                           [Final Answer : - 5 ]

  7. 65 ÷ (- 13) = ___                         [Final Answer : - 5 ]

  8. (- 3)(8) ÷ 6 = ___                        [Final Answer : - 4 ]

  9. 54 ÷ (- 3)(2) - (- 1) = ___           [Final Answer : - 54/5 ]

  10. -121 ÷ 121 = ___                        [Final Answer : - 1 ]

addition & subtraction of monomials

Adding and Subtracting Monomials

A monomial is an expression that contains a single algebraic term. For addition / subtraction of monomials, certain rules must be known:-

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Rule 1 - Must be like terms (Meaning that they must have the exact same variables and exponents).

Rule 2 - Perform the operations between the like terms and keep the variables.

Rule 3 - If the signs between the terms are the same then addition will be performed. However if the signs are different subtract and sign of the larger will be kept.

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     Example:-

                    i) 5(x)(y²) + 8(x)(y²) = 13(x)(y²)

                   As seen the variables are exactly the same, so we can take our coefficients, perform the operation and keep the variables.

                   ii) 5(y⁶) + 8(y) = 5(y⁶) + 8(y)

                   As you can see, even though both terms (monomials) have a base variable of y, they are not exactly the same. So our solution cannot be simplified any further.

                 iii) 5(y²) + 8(y²) + 3(x)(y²) 

                   Taking a look at our variables, we can combine the first two terms. This results in the following:-

                   = 13(y²) + 3(x)(y²)

Since the 2 variables are not like terms, they can not be combined and thus is our final solution.

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Simplify

Show all your Work

  1. 5(x) + 9(x) - 5(y) = ___                                              [Final Answer : 14(x) - 5(y) ]

  2. 4(x) + 9 - ( - 10) + 6(x) = ___                                    [Final Answer : 10(x) + 19 ]

  3. 3(x)(y)(z) - 5(x)(y) - 2(x)(y) - 7(x)(y)(z) =  ___      [Final Answer : - 4(x)(y)(z) - 7(x)(y) ]

  4. 14(x²) - 5(x²) + 3 - ( - 10) = ___                               [Final Answer : 9(x²) + 13 ]

  5. 15(x) - 12(z) + 9(x) - 11(z) = ___                             [Final Answer : 24(x) - 23(z) ]   

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Simplify the given expressions, then evaluate when x = -3.

Show all your Work

  1. 5(x) + 3(x) - 2(x) = ___                                           [Final Answer : -18 ]

  2. - 4(x) - (- 5(x)) = ___                                                [Final Answer : -3 ]

  3. 15(x) - (- 2(x)) + 3(x) - 10(x) =  ___                      [Final Answer : -30 ]

  4. - 2(x²) - 3(x) + 3(x²) + 5(x) = ___                          [Final Answer :  3 ]

  5. 3(x) - 5(x) - 6(x) + 8(x) = ___                                 [Final Answer : 0 ]

addition & subtraction of polynomials

Adding and Subtracting of Polynomials

A polynomial is an expression that contains multiple algebraic terms. Some of these algebraic terms, contain different powers of the same variable. For performing the operations, same rules as above apply.

     Example:-

                     i) (3x + 2) + (2x - 1) = 5x + 1

                    ii) (2x² - 7x + 3) + (x² - 5x - 1) = 3x² - 12x + 2

                   iii) (x³ + 6) + (x + 3) = x³ + x + 9

                   In all the examples above, we combined the like terms and performed the arithmetic, ending up with the final solution above.

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Simplify

Show all your Work

  1. (3x + 2) + (2x - 1) = ___                                            [Final Answer : 5x + 1 ]

  2. (y² - 2y - 6) + (3 - 2y - y²) = ___                                [Final Answer : -4y - 3 ]

  3. (x² - 2x) + (3x² + 5x) =  ___                                      [Final Answer : 4x² + 3x ]

  4. (x³ + 5x +3) + (x² - x - 1) = ___                                 [Final Answer : x³ + x² + 4x + 2  ]

  5. (y³ + 2y² + 3) + (4y² - 3y - 1) = ___                         [Final Answer : y³ + 6y² - 3y + 2  ]   

  6. (6x - 3) - (7x + y) = ___                                              [Final Answer : -x - y - 3 ]

  7. (5x + 7) - (2x + 3) = ___                                            [Final Answer : 3x + 4 ]

  8. (5x² + 6x + 1) - (2x² + x - 7) = ___                           [Final Answer : 3x² + 5x - 6 ]

  9. (x - y + 3) - (x + y + 3) = ___                                     [Final Answer : -2y ]

  10. (3x - 2) - (2x - 1) = ___                                               [Final Answer : x + 1 ]

Exponent laws

Exponent Laws

Exponents also known as powers (aⁿ) are repeated multiplication of the same entity n times. The "exponent", being n in this case, stands for the number of times a value is being multiplied. This value is known as the base.

Exponents have a few rules that need to be followed for being able to simplify mathematical expressions. There is also a table on the “General Formulae" tab, which lists all the different rules.

Here are a few examples followed by some practice problems which will help grasp this concept.

     Example:-

                     i) (x⁵)*(x⁶)

                         = (x*x*x*x*x) * (x*x*x*x*x*x)

                        As seen one could just think of it as multiplying 5 copies of x by another 6 copies or just simply using the exponent rule (x‴ )(xⁿ) = x⁽‴⁺ⁿ⁾ . This rule states that whenever you have terms with the same base, you can add the exponents.

                        = x⁽⁵⁺⁶⁾

                        = x¹¹

                   ii) (x⁵)⁶

                       = (x⁵) (x⁵) (x⁵) (x⁵) (x⁵) (x⁵)

                       =(x*x*x*x*x) (x*x*x*x*x) (x*x*x*x*x) (x*x*x*x*x) (x*x*x*x*x) (x*x*x*x*x) 

                      Just like the previous example, we can think of “to the 6th” as in multiplying 6 copies of x⁵. Whenever we have an exponent raised to a power, you can multiply both the exponent and its power together. This can be seen with the following rule (x‴)ⁿ = x‴*ⁿ.

                      = x⁽⁵*⁶⁾

                      = x³⁰

                 iii) If you have a multiple of two or more variables (x*y) or division of those (x/y) inside the brackets which is raised to a power, then the expansion of (x*y)² becomes x²*y² and for (x/y)² becomes x²/y². This rule does not work if there is a sum of difference within the brackets.

                iv) Anything raised to the power of 0 is 1.

                     Ex:- (x² + 5x + 25)⁰ = 1

                             As long as the expression is raised to the power of 0, it doesn’t matter what is inside the brackets. The final answer is always 1.

               v) All the rules that you will need are listed in the “General Formulae” tab. If you need clarification with anything please leave a query on the “Contact Us” tab

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Simplify

Show all your Work

  1. x² × x⁴ ÷ x³ = ___                                          [Final Answer : x² ]

  2. m⁴ ÷ m² × m = ___                                       [Final Answer : m³ ]

  3. x¹⁰ ÷ x³ × x⁴ = ___                                        [Final Answer : x¹¹ ]

  4. (x²)⁴ = ___                                                     [Final Answer : x⁸ ]

  5. (m²)³ ÷ m² = ___                                          [Final Answer : m⁴ ]

  6. (t³)² ÷ t⁵ = ___                                               [Final Answer : t ]

  7. x⁻³ × x⁻⁵ = ___                                             [Final Answer : x⁻⁸ ]

  8. m⁵ × m⁻⁷ = ___                                            [Final Answer : m⁻² ]

  9. (⅝)⁰ = ___                                                    [Final Answer : 1 ]

  10. (⅔)⁻³ = ___                                                  [Final Answer : ²⁷⁄₈ ]

  11. (-⅛)⁻¹ = ___                                                 [Final Answer : - 8 ] 

  12. (y³ x y⁵) ÷ y¹² = ___                                     [Final Answer : y⁻⁴ ] 

  13. (-2a⁴b⁵c⁶)⁵ = ___                                         [Final Answer : -32a²⁰b²⁵c³⁰ ]

Multiplying Monomials by monomials

Multiplying Monomials by Monomials

Example: (4x²)(2xy)(⁶⁄₄ x²y³)

                  Observation : Straight multiplication of numbers in brackets.

Step 1: Apply the associative property of multiplication to regroup like term.

             It would look like the following:-

             = (4*2*⁶⁄₄)(x²*x*x²)(y*y³)

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Step 2: Multiply the like terms together and use the product of powers to add the exponents of the variable terms as shown:

            Constants : (4*2*⁶⁄₄)                  Variables: (x⁽²⁺¹⁺²⁾)(y⁽¹⁺³⁾)

Thus the final answer is 12x⁵y⁴.

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Simplify

Show all your Work

  1. (x⁴)(x⁵) = ___                                                [Final Answer : x⁹ ]

  2. (2x²)(x³) = ___                                               [Final Answer : 2x⁵ ]

  3. (3w²)(-6w⁴) =  ___                                        [Final Answer : -18w⁶ ]

  4. (-9x⁵)(3x²)(2x⁴)(3) = ___                            [Final Answer : -162x¹¹ ]

  5. (-7x⁴y³)(x⁶y⁷)(-xy) = ___                             [Final Answer : 7x¹¹y¹¹ ]   

  6. (-7a)(-3a²b)(-4a³b³) = ___                            [Final Answer : -86a⁶b⁴ ]

  7. -9p²q³(⅓pq⁵) = ___                                      [Final Answer : -3p³q⁸ ]

  8. (2x⁴y³)⁵(-x⁴y²)³ = ___                                  [Final Answer :  -32x³²y²¹ ]

  9. 10a²b³(-2ab²)³(ab²)³ = ___                          [Final Answer : -80a⁸b¹⁵ ]

  10. (4x⁴)(3y³)(7xy) = ___                                   [Final Answer : 84x⁵y⁴ ]

Multiplying monomials by polynomials

Multiplying monomials by polynomials

Lets start with an Example:-

Expand and Simplify

(6x)(4x + 6x²y² - 8xy)

Using the distributive property of multiplication, multiply the term 6x to all the components of the polynomial (4x + 6x²y² - 8xy) as shown below.

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Solving we get the following solution: 24x² + 36x³y² - 48x²y

Expand and Simplify

Show all your Work

  1. 3(x+2) = ___                                                 [Final Answer : 3x + 6 ]

  2. -2(x - y) = ___                                                [Final Answer : -2x + 2y ]

  3. x(x - 2) = ___                                                 [Final Answer : x² - 2x ]

  4. -3x(2x - 5) = ___                                           [Final Answer : -6x² + 15 ]

  5. xy(x + y) = ___                                              [Final Answer : x²y + xy² ]

  6. 3(x + 2) + 2(x - 5) = ___                              [Final Answer : 5x - 4 ]

  7. 3(2x - 5) + 2(x + 3) + 4(x - 4) = ___           [Final Answer : 12x - 25 ]

  8. 3x(x - 2) + 5x² + 4x(x + 2) = ___                [Final Answer : 12x² + 2x ]

  9. 0.5(4x - 6y) + 0.2(10x - 15y) = ___           [Final Answer : 4x - 6y ]

  10. 3xy²(x - 3y) - 2xy²(y - 2x) = ___                  [Final Answer : 7x²y² - 11xy³ ]

Dividing monomials/Polynomials by monomials

Dividing Monomials/Polynomials By Monomials

When dividing the monomials together, divide the coefficients first followed by the variables. Firstly, regroup all the like terms using the associative property then simplify further using exponent laws.

     Example:-

                     i) (14a³b⁴) / (2ab) = 14/2(a³/a¹)(b⁴/b¹)

                                                       = 7a²b³

As observed the like terms are regrouped first and then simplified further by using quotient of powers.

When dividing a polynomial by a monomial, one can split up the problem by having each term in the numerator over the denominator (monomial). At this point, same rules for simplifying as above apply.

     Example:-

                    ii) (7x² - 14x) / (7x) = (7x²/7x) - (14x/7x)

                                                           = x - 2      

As observed, the problem was split into smaller terms (monomials over monomials). Like terms were regrouped and quotient of powers law was used to obtain the final solution.              

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Simplify

Show all your Work

  1. 6x⁶ / 3x⁶ = ___                                                                  [Final Answer : 2 ]

  2. -28s⁸ / 7s³ = ___                                                               [Final Answer : -4s⁵ ]

  3. (18x⁴y⁵) / (-2x³y) =  ___                                                  [Final Answer : -9xy⁴ ]

  4. (10x⁵y³) / (5x³y⁻¹) = ___                                                [Final Answer : 2x²y⁴ ]

  5. (-18xyz) / (-27x²y²z⁻¹) = ___                                          [Final Answer : 2z²/3xy ]   

  6. (3x + 3y) / 3 = ___                                                            [Final Answer : x + y ]

  7. (4y² - 4) / 4 = ___                                                              [Final Answer : y² - 1 = (y + 1)(y - 1) ]

  8. (5x²y - 10xy²) / (5xy) = ___                                            [Final Answer : x - 2y ]

  9. (20m³n² - 10mn³ - 30m²n²) / (-5mn) = ___                  [Final Answer : -4m²n + 2n² + 6mn ]

  10. (18a³b²c - 12a²b³c + 24a²b²c²) / (6a²b²c²) = ___      [Final Answer : 3ac⁻¹ - 2bc⁻¹ + 4 ]

Common Factoring

Common Factoring

Common factoring is one of the easiest type of factoring. As discussed, monomials are expressions that contain a single algebraic term. Polynomials are terms that contain many of these algebraic terms (monomials). Monomial factors are factors which have numbers/variables to some exponent. Finding this common monomial means finding the common factor from the given set of terms. The common monomial factor can be a combination of variables and numbers.

 

Main step to factor is to find the common factor whether it’s a number/variable. Refer to the formulae listed in the “General Formulae” tab. 

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     Example: Factor

                      5x² + 10x

Observe that 5x is the greatest common factor of 5x² and 10x. Hence we conclude that the final answer is 5x(x+2).

When factoring, always look for the greatest common factor. This is needed to factor fully.

 

Factor

Show all your Work

  1. 7xy - 14x = ___                                        [Final Answer : 7x(y - 2) ]

  2. 8y² - 12y = ___                                         [Final Answer : 4y(2y - 3) ]

  3. 6x² - 9x = ___                                           [Final Answer : 3x(2x - 3) ]

  4. 5z³ + 10z² = ___                                      [Final Answer : 5z²(x + 2) ]

  5. 7x²yz + 14yz³ = ___                                [Final Answer : 7yz(x² + 2z²) ]

  6. 2xy - 6x + y - 3 = ___                               [Final Answer : (2x + 1)(y - 3) ]

  7. 8x²y² - 12xy = ___                                   [Final Answer : 4xy(2xy - 3) ]

  8. 24d² + 40d + 56 = ___                           [Final Answer : 8(3d² + 5d + 7) ]

  9. 9x² + 6x - 12 = ___                                  [Final Answer : 3(3x² + 2x - 4) ]

  10. 16y² - 32y + 6 = ___                               [Final Answer : 2(8y² - 16y + 3) ]

Product of binomials

Product of Binomials

A binomial is an algebraic expression which is either a sum or difference of two terms. When multiplying two binomials the FOIL rule applies:-

F – First (First term of each binomial are multiplied together)

O – Outer (First term of the first binomial and the second term of the second are multiplied together)

I – Inner (Second term of the first binomial and the first term of the second are multiplied together)

L – Last (Second term of each binomial are multiplied together)

 

Binomial General Form – (a+b)(c+d) = ac + ad + bc + bd

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Example: Expand and simplify

                 (3x + 2)(x + 4)

Following the foil principle we end up with 3x² + 12x + 2x + 8. Simplifying this further we end up with our final solution:  3x² + 14x + 8.

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Expand and Simplify

Show all your Work

  1. (x + 2)(x + 5) = ___                                                 [Final Answer : x² + 7x + 10 ]

  2. (x + 1)(x + 2) = ___                                                 [Final Answer : x² + 3x + 2 ]

  3. (x - 4)(x - 2) = ___                                                    [Final Answer : x² - 6x + 8 ]

  4. (p - 4)(p + 1) = ___                                                  [Final Answer : p² - 3p - 4 ]

  5. (q + 3)(q - 7) = ___                                                  [Final Answer : q² - 4p - 21 ]

  6. (3x + 1) (2x + 3) = ___                                           [Final Answer : 9x² + 6x + 1 ]

  7. (3x + 5)(2x - 7) = ___                                             [Final Answer : 6x² - 11x - 35 ]

  8. (3y - 8)(2y + 5) = ___                                             [Final Answer : 6y² - y - 40 ]

  9. (2x + 7)(3x - 5) = ___                                             [Final Answer : 6x² + 11x - 35 ]

  10. 0.5(3x + 2)(4x + 6) = ___                                      [Final Answer : 6x² + 13x + 6 ]

Factoring Trinomials

Factoring Trinomials

A trinomial is an algebraic expression that consists of 3 terms. Lets begin with an example:-

    

     Example: Factor Fully

                     i) 5x² + 15x + 10

As seen, the above example follows the form ax² + bx + c. In this case, this trinomial can be further simplified into the form x² + bx + c. First find the greatest common factor (GCF). Once found, factor out this GCF and you are left with the following:-

                    = 5(x² + 3x + 2)

Now to factor "x² + 3x + 2", the product must have the form (x + _)(x + _), where the numbers in the "_" must be factor of 2 and must add up to the middle term of 3x. Therefore the only factors that would satisfy this are 2, 1. 

Therefore    5(x² + 2x + 1x + 6)

 Now applying the factor by grouping method we get:-

                    =5[x(x + 2) + 1(x + 6)]

                    =5[(x + 1)(x + 6)]

Thus our final solution is 5[(x + 1)(x + 6)].

​

                   ii) 6x² + 11x - 10

As seen, the above example follows the form ax² + bx + c.

Now to factor "6x² + 11x - 10" the product must have the form (x + _)(x - _), where the numbers in the "_" must be factor of -60 (product of the first and last numbers) and must add up to the middle term of 11x. Therefore the only factor combination that would satisfy this are +15, -4. 

Therefore   6x² +15x - 4x - 10

 Now applying the factor by grouping method we get:-

                    =[3x(2x + 5) - 2(2x + 5)]

                    =[(3x - 2)(2x + 5)]

Thus our final solution is (3x - 2)(2x + 5).

​

Expand and Simplify

Show all your Work 

  1. x² - x - 2 = ___                                                [Final Answer : (x - 2)(x + 1) ]

  2. y² + y - 30 = ___                                            [Final Answer : (x - 5)(x + 6) ]

  3. 3y² - 21y + 36 = ___                                     [Final Answer : (x - 4)(x - 3) ]

  4. 5x² - 35x + 30 = ___                                     [Final Answer : (x - 6)(x - 1) ]

  5. 2x⁴ + 16x³ + 24x² = ___                              [Final Answer : 2x²(x + 6)(x + 2) ]

  6. 9x² + 19x + 10 = ___                                    [Final Answer : (x + 1)(9x + 10) ]

  7. 2x² - 11x + 12  = ___                                    [Final Answer : (x - 4)(2x - 3) ]

  8. 15x² + 26x + 8  = ___                                   [Final Answer : (5x + 2)(3x + 4) ]

  9. 36x² + 66x - 60  = ___                                  [Final Answer : 6(3x - 2)(2x + 5) ]

  10. 24x² - 68x - 180 = ___                                  [Final Answer : 4(2x - 9)(3x + 5) ]

Factoring a difference of squares

Factoring a Difference of Squares

A difference of squares is a polynomial which means subtraction of the square of one term from the square of another and is expressed as (a² - b²). Difference of squares property is the  following:-

                                                              a² - b² = (a + b)(a - b)

​

Difference of the squares of a & b, results in the product of the sum and difference of a and b. The above property is also listed in the “general formulae” tab for more information.

Lets start with an example:-

     Example: Factor

                     i) x² - 4 = (x + 2)(x - 2)

Following the above property, this can be written as the following: (x)² - (2)². Thus the final solution using the difference of squares property is (x + 2) (x - 2).

​

                    ii) 8m² - 2n² = 2(4m² - n²) = 2(2m + n)(2m - n)

Note always simplify the problem as much as possible before factoring. Thus, above can be simplified to 2(4m² - n²). This can be written as 2[(2m)² - (n)²]. Therefore, the final solution using the difference of squares property is 2(2m + n)(2m - n).

            

Factor

Show all your Work 

  1. x² - 9 = ___                                                [Final Answer : (x - 3)(x + 3) ]

  2. y² - 16 = ___                                             [Final Answer : (y - 4)(y + 4) ]

  3. 4d² - 36 = ___                                          [Final Answer : 4(d + 3)(d - 3) ]

  4. 81 - 121y² = ___                                      [Final Answer : (9 + 11y)(9 - 11y) ]

  5. 100x² - 1 = ___                                        [Final Answer : (10x + 1)(10x - 1) ]

  6. x²y² - 1 = ___                                            [Final Answer : (xy + 1)(xy - 1) ]

  7. 121w² - 49t²  = ___                                 [Final Answer : (11w + 7t)(11w - 7t) ]

  8. (a + 2)² - b²  = ___                                   [Final Answer : ((a + 2) + b)((a + 2) - b) ]

  9. (2p - 3)² + q²  = ___                                 [Final Answer : ((2p - 3) + q)((2p - 3) - q) ]

  10. (y + 2)² - 25 = ___                                   [Final Answer : ((y + 2) + 5)((y + 2) - 5) ]

Product of polynomials

Product of Polynomials

Recap:-

  • A monomial is an expression that contains a single algebraic term.

  • A polynomial is an expression that contains multiple algebraic terms.

  • A binomial is an algebraic expression (polynomial) which is either a sum or difference of two terms. When multiplying two binomials the FOIL rule applies.

  • A trinomial is an algebraic expression (polynomial) that consists of 3 terms.

 

Lets start with an example:-

     Example: Multiply

                     (x² - 10x + 25)(x + 5)

Note that FOIL won't work here, because there are more than two terms now. Always remember to multiply each term in the first polynomial by each term in the second polynomial and add the like terms together to get to the final solution.

Following the stated procedure, we end up with the following: x³ + 5x² - 10x² - 50x + 25x + 125. Simplifying this further by adding all the like terms together we get x³ - 5x² - 25x + 125.

            

Expand and Simplify

Show all your Work 

  1. (x + 3)(x + 1) = ___                                              [Final Answer : x² + 4x + 3 ]

  2. (3x + 5)(4x + 3) = ___                                         [Final Answer : 12x² + 29x +15 ]

  3. (x² - 5x + 2)(x - 2) = ___                                      [Final Answer : x³ - 7x² + 12x - 4 ]

  4. (2x² - 7x + 1)(x + 4) = ___                                  [Final Answer : 2x³ + x² - 27x + 4 ]

  5. (2x² + x - 2)(3x - 5) = ___                                    [Final Answer : 6x³ - 7x² - 11x + 10 ]

  6. (x² - 5x + 3)(x² + x - 2) = ___                              [Final Answer : x⁴ - 4x³ - 4x² + 13x - 6 ]

  7. (2x² - 7x + 5)(3x² + x - 2) = ___                         [Final Answer : 6x⁴ - 19x³ + 4x² + 19x - 10 ]

  8. (3x² - 8x + 1)(2x² - 5) = ___                                [Final Answer : 6x⁴ - 16x³ - 13x² + 40x - 5 ]

  9. (3x + 2y)² - (2x + 3y)²  = ___                              [Final Answer : 5x² - 5y² ]

  10. 3(2x - 3)² - 2(3x + 1)² = ___                                [Final Answer : - 6x² - 48x + 25 ]

Solving equations by Addition/subtraction,Division & Multiplication

Solving Equations by Addition/Subtraction, Division & Multiplication

When solving equations, one must find a value for a variable that makes the equation true. From all the properties that we have learned, one can rearrange the terms of an equation and find the value for the variable. One can also think of this as balancing a weigh scale, so whatever operations are performed on one side, must be done on the other side of the equation as well.

 

     Example:

                    i) x + 5 = 10

To solve, one must isolate for the variable x. In this case one must move 5 to the other side of the equation. To do this, we are going to follow the addition property of equality. It states that if we add a number to one side of an equation, we must add the same number to the other side to balance.

Following the rules we get x + 5 + (- 5) = 10 + (- 5). Simplifying we end up with the final solution of  x = 5.

​

                   ii) x - 26 = 34

Same as before to solve, one must isolate for the variable x. In this case one must move 26 to the other side of the equation. To do this, we are going to follow the addition property of equality. It states that if we add a number to one side of an equation, to satisfy we must add the same number to the other side. Following the rules we get x - 26 + (26) = 34 + (26). Simplifying we end up with the final solution of x = 60.

    

                 iii) 5x = 30

In this case one must move 5 to the other side of the equation. To do this, we must use inverse operations and divide both sides by the same number to find the value of x. Following the rules we get 5x/5 = 30/5. Simplifying we end up with the final solution of x = 6.

    

                 iv) x/2 = 7

In this case one must move 2 to the other side of the equation. To do this, we must use inverse operations and multiply both sides by the same number to satisfy the balance and find the value of the variable ‘x’. Following the rules we get 2(x/2) = 7*2. Simplifying we end up with the final solution of x = 14.

​

We can also check our solutions by substituting the value of the variable back in. If the LHS equals the RHS of the equation, the value of the variable is correct.

​

Solve the following.

Show all your Work 

  1. x + 5 = 7                                                [Final Answer : x = 2 ]

  2. x - 4 = 11                                               [Final Answer : x = 15 ]

  3. 5 + y = 25                                              [Final Answer : y = 20 ]

  4. - 16 + 12x = - 11 + 11x                        [Final Answer : x = 5 ]

  5. 75 + 8x = - 13 + 7x                               [Final Answer : x = - 88 ]

  6. - 63 + 15s = 14s + 18                          [Final Answer : s = 81 ]

  7. 3x = 15                                                  [Final Answer : x = 5 ]

  8. - 39 = - 13x                                            [Final Answer : x = 3 ]

  9. 3x + 12 = 39                                         [Final Answer : x = 9 ]

  10. 25 + 5x = 135                                       [Final Answer : x = 22 ]

  11. 7x - 14 = 5x + 42                                  [Final Answer : x = 28 ]

  12. 2(x + 4) = 7(x - 3) + 34                        [Final Answer : x = - 1 ]

  13. 0.5x = 8                                                  [Final Answer : x = 16 ]

  14. s/6 = 0                                                    [Final Answer : s = 0 ]

  15. (2x + 3) / 4 = 9/4                                 [Final Answer : x = 3 ]

  16. x/2 = x/3 + 5                                         [Final Answer : x = 30 ]

  17. x/6 + 1 = x/7                                         [Final Answer : x = - 42 ]

  18. (x - 1) / 4 = (3x + 2) / 6                       [Final Answer : x = - 7/3 ]

Solving Inequalities

Solving Inequalties

Inequalities are ways of comparing 2 numbers using inequality symbols. The symbols used are:-

  1. Less than - <

  2. less than or equal to - ≤

  3. greater than - >

  4. greater than or equal to - ≥

When solving inequalities (just as solving equations), find value for a variable that makes the inequality true. One can rearrange the terms of an inequality and find the solution. The operations performed on one side, must be done on the other side of the inequality as well. Also if possible, always try to simplify the problem. This can be done by flipping the inequality if the variable is not already on the left side. Solutions can also be graphed to verify the solution set.

​

Solving Inequalities by Addition and Subtraction

     Example:

                     i) x + 12 < 15

To solve, one must isolate the variable x. The operations performed on one side must be done to the other as well. In this case one must add -12 to both sides.

Solving, we get x + 12 + (- 12) < 15 + (- 12). Simplifying we end up with the final solution of x < 3.

Graphing the solution we get...

Solving Inequalities by Addition and Subtraction

Note the number 3 is marked by an open circle as its not included in the solution. The solution includes all the values that are less than 3.

                    

                    ii) x - 10 ≤ -13

To isolate the variable x one must add +10 on both sides.  Simplifying we end up with the final solution of x ≤ -3.

Note the number -3 is marked by a filled circle as it is included in the solution. The solution includes all the values that are less than or equal to -3.

 

Solving Inequalities by Multiplication/Division

Similar rules as before apply. The only exception is that when an inequality is multiplied or divided by a negative, the direction of the inequality changes.

     Example:

                     iii) x/6 < -9

To isolate for x, we must use inverse operations and multiply both sides by the same number. Solving and simplifying, we get 6(x/6) < -9*6 which results to our final solution of x < -54.

Graphing the solution we get...

Solving Inequalities by Multiplication/Division

Note the number -54 is marked by an open circle as its not included in the solution. The solution includes all the values that are less than -54.

​

                     iv) 30 < -6x

From observation it is seen that the variable x is on the RHS of the inequality. To simplify, flip the inequality such that the variable x is on the LHS (left hand side).

It becomes:     -6x > 30

Now to isolate for x, we must use inverse operations and divide both sides by the same number. Solving and simplifying, we get (-6x)/-6 > -9/-6 which results to our final solution of x < 3/2.

Graphing the solution we get...

Note the number 1.5 is marked by an open circle as its not included in the solution. The solution includes all the values that are less than 1.5.

 

                     v) 3(x – 2) > 2x + 5

To solve, expand and simplify the the LHS.

It should be as follows:   3x – 6 > 2x + 5

Through further simplification, our final solution is x > 11 .

Graphing the solution we get...

Solve and Graph the solution set. Variables have domain R.

Show all your Work 

  1. x + 3 > 7                                                                 [Final Answer : x >4 ]

  2. x + 9 < 2                                                                 [Final Answer : x < -6 ]

  3. x - 1 ≤ -5                                                                 [Final Answer : x ≤ -4 ]

  4. -7 + x < -6                                                               [Final Answer : x < 1 ]

  5. 5x - 4 < 4x + 3                                                       [Final Answer : x < 7 ]

  6. -11 + 9w > 8w - 31                                               [Final Answer : w > -20 ]

  7. 3(x - 2) > 2x + 5                                                    [Final Answer : x > 11 ]

  8. 3(2x - 1) > 5x - 16                                                 [Final Answer : x > -13 ]

  9. 13x - 21 ≤ 3(2x - 1) + 6x                                     [Final Answer : x ≤ 18 ]

  10. 5x - 7.4 > 4x - 8.5                                                 [Final Answer : x > -1.1 ]

  11. 2x > 6                                                                     [Final Answer : x > 3 ]

  12. 4t ≤ 20                                                                   [Final Answer : t ≤ 5 ]

  13. -2x > -8                                                                   [Final Answer : x < 4]

  14. 5x + 3 ≤ 3x + 7                                      [Final Answer : x ≤ 2 ]

  15. 4(x + 1) - 2(x + 3) ≥ 4(x - 1)                                [Final Answer : x ≤ 1 ]

  16. x/2 > 8                                                                    [Final Answer : x > 16 ]

  17. -x/2 ≥ -6                                                                 [Final Answer : x ≤ 12 ]

  18. -t/5 ≥ -1                                                                  [Final Answer : t ≤ 5 ]

  19. (1/2)x + 3 ≥ (3/4)x - 1                                         [Final Answer : x ≤ 16 ]

  20. (x + 1)/3 > (x - 2)/4                                              [Final Answer : x > -10 ]

Ratio And Rate

Ratio and Rate

A ratio is a relationship or comparison between two numbers by division. For example if a grocery store sells 3 apples for every 2 oranges they sell. The ratio of apples to oranges would be 3 to 2, 3:2. This ratio can also be expressed as a fraction or decimal, 3/2 in decimal form 1.5.

A rate is type of ratio that involves comparing different units. A unit rate is a ratio that takes time into consideration. Unit rates usually have a denominator of 1 for simplicity of comparing other rates.           

Some common examples of unit rates:-

  • 40 m/s => 40 meters / 1 second

  • 60 m/h => 60 miles / 1 hour

​

Here are a few examples:

1. Express the following ratios in simplest form.

    i) 4:2 => 4/2 = 2/1 => 2:1

   ii) 8:12 => 8/12 = 2/3 => 2:3

  iii) 15 cm to 35 cm => 15/35 = 3/7 => 3:7

    As observed above think of it as simplifying fractions.

​

2. Express each of the following rates as the rate for 1 unit.

     i) 480m in 5s = 480m / 5s = 96m/s

    ii) 862km in 2.25hrs = 862km / 2.25hrs = 383.11 km/hr

    Note the final answer is in terms of per unit time.

​

Solve the following.

Show all your Work 

  1. Express the following ratios in simplest form.

  • 3:9                                                                      [Final Answer: 1:3]

  • 15:10                                                                 [Final Answer: 3:2]

  • 48:16                                                                 [Final Answer: 3:1]

  • 8:44                                                                   [Final Answer: 2:11]

  • 90,000 m² to 4 ha                                           [Final Answer: 9:4]

  2. State an equivalent ratio for each of the following.

  • 1:5                                                                      [Final Answer: 2:10]

  • 7:1                                                                      [Final Answer: 14:2]

  • 40:90                                                                 [Final Answer: 4:9]

  • 7:5                                                                      [Final Answer: 21:15]

  • 1:10                                                                   [Final Answer: 4:40]

  3. Express each of the following rates as the rate for 1 unit.

  • 480 m in 5 s                                                      [Final Answer: 96 m/s]

  • 72 kg for $2.40                                                [Final Answer: 30 kg/$]

  • 512 words in 6 min                                          [Final Answer: 85.3 w.p.m]

  • 65 cm in 2.5 min                                              [Final Answer: 26 cm/min]

  • $250.00 for 20.00 hrs                                     [Final Answer: 12.5 $/hr]

  4. Which is the better value?

  • 750 mL for $1.29 or 500 mL for $0.85

          [Final Answer: 500 mL for $0.85]

  • 20 kg for $24 or 50kg for $65

         [Final Answer: 20 kg for $24 ]

  5. Calculate the unit price for each of the following.

  • 550 mL of pop for $1.50                                 [Final Answer: 2.73 $/L]            

  • 284 g of chocolate for $ 2.25                         [Final Answer: 7.92 $/kg]

Pythagorean Theorem

The pythagorean theorem

According to Pythagoras’s theorem, an equation for any right triangle (as above) can be derived into the following form:-

C² = A² + B²

Where:-

  • C represents the longest side (hypotenuse) of the triangle.

  • A, B represents the lengths of the other 2 sides.

This relates the lengths of all the 3 sides of any right angle triangle. Thus the square of the hypotenuse results in the sum of the squares of the other 2 sides.

    Example: Find the length of the unknown side.

i)

Step 1: Write down the Pythagorean Theorem.
            C² = A² + B²

Step 2: Substitute in all the given values. In this case for A, B.

            C² = 3² + 4²

Step 3: Simplify and Solve

            C² = √25

            C = 5

Thus, the hypotenuse c is 5.

ii)

Step 1: Write down the Pythagorean Theorem.

           C² = A² + B²

Step 2: Substitute in all the given values.

           24² = 13² + r²

Step 3: Simplify and Solve

            r² = 24² - 13²

            r² = √407

            r = 20.2

Thus, the side r is equal to 20.2

​

Solve the following.

Show all your Work 

  1.

Answer: x = 10.6

  2.

Answer: z = 24.1

  3.

Answer: y = 14.28

  4.

Answer: x = 7

  5.

Answer: y = 21.2

  6.  A table top is 30 inches wide and 60 inches long. How long is its diagonal?

     Answer: Thus the diagonal of the table top would be 67.1 inches.

​

  7.  A 11 foot ladder is leaning against a mast on a ship. If the ladder is 3 feet away from the base of the mast, how high up on the mast will it go?

     Answer: Thus the ladder will reach a height of 10.6 ft on the mast.

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