Entry Task What is the polynomial function in

Entry Task • What is the polynomial function in standard form with the zeros of 0, 2, -3 and -1?

5. 3 Solving Polynomial Equations Learning Target: Students will be able to understand how to solve polynomial equations by factoring

Factoring out the GCF Factoring a polynomial with a common monomial factor (using GCF) Always look for a GCF before using any other factoring method.

Steps: 1. Find the greatest common factor (GCF). 2. Divide the polynomial by the GCF. The quotient is the other factor. 3. Express the polynomial as the product of the quotient and the GCF.

5 12 x – 3 18 x – 2 3 x

Difference of Squares To factor, express each term as a square of a monomial then apply the rule. . .

4 Steps for factoring Difference of Squares 1. Are there only 2 terms? 2. Is the first term a perfect square? 3. Is the last term a perfect square? 4. Is there subtraction (difference) in the problem? If all of these are true, you can factor using this method!!!

1. Factor 2 x - 25 When factoring, use your factoring table. Do you have a GCF? No Are the Difference of Squares steps true? Two terms? Yes 2 – 25 x 1 st term a perfect square? Yes 2 nd term a perfect square? Yes Subtraction? Yes Write your answer! ( x + 5)( x - 5)

2. Factor 2 16 x -9 When factoring, use your factoring table. Do you have a GCF? No Are the Difference of Squares steps true? Two terms? Yes 2– 9 16 x 1 st term a perfect square? Yes 2 nd term a perfect square? Yes Subtraction? Yes Write your answer! (4 x + 3)(4 x - 3)

3. Factor 2 81 a – 2 49 b When factoring, use your factoring table. Do you have a GCF? No Are the Difference of Squares steps true? Two terms? Yes 2 – 49 b 2 81 a 1 st term a perfect square? Yes 2 nd term a perfect square? Yes Subtraction? Yes Write your answer! (9 a + 7 b)(9 a - 7 b )

Try these on your own:

Perfect Square Trinomials • can be factored just like other trinomials but if you recognize the perfect squares pattern, follow the formula!

a b Does the middle term fit the pattern, 2 ab? Yes, the factors are (a + b)2 :

a b Does the middle term fit the pattern, 2 ab? Yes, the factors are (a - b)2 :

1) Factor x 2 + 6 x + 9 Does this fit the form of our Perfect Square Trinomials perfect square trinomial? (a + b) = a + 2 ab + b (a - b) = a – 2 ab + b 1) Is the first term a perfect square? Yes, a = x Since all three are true, 2) Is the last term a perfect write your answer! square? 2 (x + 3) Yes, b = 3 3) Is the middle term twice the product of the a and b? You can still factor the other way but this Yes, 2 ab = 2(x)(3) = 6 x 2 2 2 is quicker!

2) Factor y 2 – 16 y + 64 Does this fit the form of our Perfect Square Trinomials perfect square trinomial? (a + b) = a + 2 ab + b (a - b) = a – 2 ab + b 1) Is the first term a perfect square? Yes, a = y Since all three are true, 2) Is the last term a perfect write your answer! square? 2 (y – 8) Yes, b = 8 3) Is the middle term twice the product of the a and b? Yes, 2 ab = 2(y)(8) = 16 y 2 2 2

Sum and Difference of Cubes:

Write each monomial as a cube and apply either of the rules. Rewrite as cubes Apply the rule for sum of cubes:

Rewrite as cubes Apply the rule for difference of cubes:

CUBIC FACTORING EX- factor and solve 8 x³ - 27 = 0 a³ - b³ = (a - b)(a² + ab + b²) 8 x³ - 27 = (2 x - 3)((2 x)² + (2 x)3 + 3²) (2 x - 3)(4 x² + 6 x + 9)=0 X= 3/2 Quadratic Formula

Factoring By Grouping (use with 4 or more terms) 1. Group the first set of terms and last set of terms with parentheses. 2. Factor out the GCF from each group so that both sets of parentheses contain the same factors. 3. Factor out the GCF again (the GCF is the factor from step 2).

Example : 3 b – 2 3 b + 4 b - 12 Step 1: Group Step 2: Factor out GCF from each group Step 3: Factor out GCF again

Example :

Try these on your own:

Answers:

Remember…These are the methods from this section

Homework p. 301 #15 -35 odds Challenge - #53