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.

12x 5 – 18x 3 – 3x 2

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

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

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

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

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

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

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

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

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

Sum and Difference of Cubes :

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

Rewrite as cubes Apply the rule for difference of cubes:

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

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).

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

Example :

Try these on your own:

Answers:

Remember…These are the methods from this section

Homework p. 301 #5-35 odds Challenge - #53