MCF3M UNIT 2 - FACTOR QUADRATIC EXPRESSIONS
Prerequisite Skills
Expand and Simplify Algebraic Expressions
* Order of Operations: BEDMAS
* To add/subtract monomials, you “Collect Like Terms”.
* To multiply/divide monomials, you use the Exponent Laws.
1) Multiplication Law :
2)Division Law :
3)Power Law :
- DISTRIBUTIVE PROPERTY
- Used to eliminate Brackets:a(b + c) = ab + ac
- To multiply Binomials (FOIL)(a + b)(c + d) = ac + ad + bc + dc
Examples : Simplify
a) 3x – 5y + 6x b) 4m2n – 3mn2 +6m2n + mn2c) 3(2x - 4)
d) -4(2x2+ 7x +3)e) f) (x + 4)(x – 6)
g) (2x – 5)(x + 1)g) y + 2 – (y - 4)(y – 5) h) 7(x + 6) + 6
Factors of Polynomials
Greatest Common Factor – The greatest number and/or variable that is a factor of two or more numbers or terms.
Ex. The GCF of 8 and 12 is 4. The GCF of 6x2y and 12xy2 is 6xy.
Factoring is the opposite to expanding
Factor: 2x – 4Expand: 2(x – 2)
= 2(x – 2)= 2x - 4
Factoring by the GCF
- Determine the GCF and place in front of a bracket.
- Divide each term by the GCF to determine the terms leftover in the bracket.
Example: Factor the following.
a) 2xy + 6yb) 3m2 n3 – 9m5 n5 + 27m3 n7c) 2x2 + 7x
d) 12p2 – 4pe) 9n2 + 15n – 18e) 5(x – 4) + 2(x – 9)
2.1 QUADRATIC FUNCTIONS: EXPLORING FORMS
There are 3 forms used to model quadratic functions.
FORM / MODEL / PROPERTIES / EXAMPLEStandard Form / y = ax2+ bx + c
where a,b & c are constants and a≠0 /
- If a › 0, the parabola opens
- If a ‹ 0, the parabola opens down and has a minimum.
- c’ is the y-intercept
a = 3 and 3 › 0, so the parabola opens up and has a minimum.
7 is the y-intercept.
Factored Form / y = a(x – r)(x – s)
where a, r & s are constants and a≠0. /
- If a › 0, the parabola opens
- If a ‹ 0, the parabola opens down and has a minimum.
- Values for r and s are the x-intercepts or zeros.
a= -2 and -2 ‹ 0, so the parabola opens down and has a maximum.
The x-intercepts are at -4 and 3.
Vertex Form / y = a(x – h)2 + k
where a, h and k are constants and a≠0. /
- If a › 0, the parabola opens
- If a ‹ 0, the parabola opens down and has a minimum.
- (h, k) is the vertex
a = 0.5 and 0.5›0 so the parabola opens up and has a minimum.
The vertex is at (3,5).
Increasing and Decreasing Functions
- A function is increasing when » as x increases, y also increases.
- A function is decreasing when as x increases, y decreases.
Quadratic Functions have are both increasing and decreasing functions. They increase on certain intervals and decrease on others.
Example:
Example: For the quadratic function y = -2x2 + 6x – 3, identify
i)the form of the function
ii)the direction of opening
iii)whether the vertex is a minimum or a maximum
iv)the y-intercept
Example: For the quadratic function , identify
i)the form of the function
ii)the direction of opening
iii)the x-intercepts
iv)the vertex and whether it is a max or min
v)the y-intercept
vi)the intervals for which the function is increasing or decreasing.
Example: For the quadratic function , identify
i)the form of the function
ii)the direction of opening
iii)the vertex and whether it is a max or a min
iv)the axis if symmetry
v)the y-intercept
vi)the intervals for which the function is increasing or decreasing.
2.2QUADRATIC FUNCTIONS: COMPARING FORMS
Simplify Quadratic Expressions
Recall: Basic Distributive Property: a(b + c) = ab + ac
Distributive Property for Multiplying Binomials (FOIL): (a + b)(c + d) = ac + ad +bc + bd
Example: Simplify the following quadratic expressions
a) (y + 3)(y – 6) b) (3x + 2)2c) (2n – 1)(n + 1) + (n + 2)2
Converting from Factored Form to Standard Form
- Expand the quadratic expression using the distributive property.
- Collect like terms.
Example: Write each quadratic function in standard form, identify the x-intercepts, y-intercept and state the direction of opening. Sketch a graph.
a) f(x) = -(x + 4)(x - 3)b)
Converting from Vertex Form to Standard Form
- Expand the expression using the distributive property.
- Collect like terms.
Example; Rewrite the quadratic function in standard form, state the coordinates of the vertex, direction of opening and the y-intercept. Sketch a graph.
2.3FACTOR QUADRATIC EXPRESSIONS OF THE FORM ax2 + bx + c
Recall : Factoring is the opposite of expanding.
Ex. Expand 2x(x + 5) = 2x2 + 10x Factor 2x2 + 10x = 2x(x + 5)
* You have already be familiar with “Greatest Common Factor” or GCF factoring.
The Sum and Product Method:
1) Multiply the a and c value together. This is the product.
2) The middle coefficient, b, is the sum.
3) Find 2 numbers that multiply to give you the product (ac) and also add up to give you the sum (b value). Make sure these 2 numbers have the correct signs.
4) Make two fractions. The denominators are the 2 numbers you found in step 3. Include the signs in the denominator. The numerators of both fractions are ax.
5) Reduce the fractions to lowest terms where possible.
6) The factors are two binomials ( )( ). The factors can be filled in by reading the fractions.
7) Check by expanding.
Examples:
1. Factor the following expressions. Then check by expanding.
a) x2 + 9x + 20 b) 2a2 + 5a – 3
c) m2 – 11m + 30 d) 4a2 – 3ab – 10b2
* Remember to always factor out a GCF first if possible!
2. Factor fully. Then check by expanding.
a) 2x2 + 22x + 56 b) 40m2 + 170mn + 40n2
Determine the X-Intercepts of a Quadratic Function by Factoring
Example: For the following quadratic functions,
i)express the function in its factored form.
ii)find the x-intercepts
iii)find the coordinates of the vertx and the axis of symmetry
iv)sketch a graph
a) f(x) = x2 – 4x – 12b) f(x) = 2x2 – 8x + 6
Not every quadratic function has a factored form.
For example: f(x) = -2(x – 3)2 - 1 has a vertex at (3, -1) and it opens downward.
2.4SELECT AND APPLY FACTORING STRATEGIES
Perfect Square Trinomials
- A perfect square trinomial will have one of the following forms
Expanded FormFactored Form
p2x2 + 2pqx + q2 = (px + q)2
p2x2 - 2pqx + q2 = (px - q)2
Examples: Factor the following.
a) x2 – 6x + 9b) 25x2 + 20x + 4c) 3x2 – 24x + 48
Difference of Squares
A “Difference of Squares” is a binomial with a subtraction sign between the two terms and both terms are perfect squares.
- A difference of squares will have one of the following forms
Expanded FormFactored Form
p2x2 – q2= (px + q)(px – q)
q2 – p2x2= (q + px)(q – px)
* To factor a “Difference of Squares” :
1) Set up the factored form: 2 binomial factors : ( )( )
2) Place the of the first term at the beginning of both brackets.
3) Place the of the second term at the end of both brackets.
4) Put a plus sign in one bracket and a minus sign in the other bracket.
5) Check by expanding. (Optional)
Examples :
1. Factor the following expressions. Then check by expanding.
a) x2 – 16 b) 25 – m2 d) 9n2 – 1
***** When factoring, always factor out a GCFfirst if possible!*****
2. Factor fully. Then check by expanding.
a) 2x2 – 50 b) 12a2 – 75 c) 4m3 – 64m
Apply factoring to Analyze Quadratic Functions
Example: For the following functions determine their
i)factored form
ii)x-intercepts
iii)vertex
iv)direction of opening
a) f(x) = x2 – 4x + 4
b) f(x) = 2x2 – 18
c) f(x) = -3x2 -6x + 72
2.5SOLVE QUADRATIC EQUATIONS BY FACTORING
When two factors have a product of ZERO, either one or both of the factors must be equal to ZERO.
If A x B = 0, then either A = 0 or B = 0 or both = 0.
- Solutions to a quadratic equation are often called the roots of the equation.
- Steps to solving quadratic equations :
1)Ensure the equation is in the form ax2 + bx + c = 0.
2)Factor the equation.
3)Set each factor equal to zero and solve for x. These are the roots of the equation.
Examples :
- Find the roots.
a) 3x2 – 12x = 0 b) x2 – 4x – 21 = 0
c) a2 – 64 = 0d) 4m2 + 18m + 8 = 0
d) 9c2 – 12c = 0e) 8n2 – 18 = 0
f) 4p2 – 20p + 29 = 4
Find and Interpret the Zeros of a Quadratic Function
Examples
1) A person standing on a bridge tosses a rock into the water 20m below. During its flight, the rock reaches a maximum height 1.5 seconds after it has been thrown, and enters the water after 4 seconds.
a)Draw a diagram and label important points.
b)Determine the zeros.
c)Explain the meaning of each zero.
d)Determine the equation that models this scenario.
e)Determine the maximum height reached by the rock.
2) A ball is thrown from a cliff. The path of the ball is modeled by the equation h = -5t2+ 5t + 210, where h is the height, in meters, of the ball above the ground and t is the time, in seconds, after it is thrown. Find the zeros of the function and explain their significance.