Factoring Polynomials

Factor the following:  

A “Difference of Squares” is a binomial (. 2 terms only A “Difference of Squares” is a binomial (*2 terms only*) and it factors like this:

Factoring a polynomial means expressing it as a product of other polynomials.

Factoring Method #1 Factoring polynomials 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.

Step 1: Step 2: Divide by GCF

The answer should look like this:

Factor these on your own looking for a GCF.

Factoring polynomials that are a difference of squares. Factoring Method #2 Factoring polynomials that are a difference of squares.

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

Here is another example:

Try these on your own:

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:

Factoring Method #3 Factoring a trinomial in the form:

2. Product of first terms of both binomials Factoring a trinomial: 1. Write two sets of parenthesis, ( )( ). These will be the factors of the trinomial. 2. Product of first terms of both binomials must equal first term of the trinomial. Next

Factoring a trinomial: 3. The product of last terms of both binomials must equal last term of the trinomial (c). 4. Think of the FOIL method of multiplying binomials, the sum of the outer and the inner products must equal the middle term (bx).

x -2 -4 O + I = bx ? Factors of +8: 1 & 8 2 & 4 -1 & -8 -2 & -4 1x + 8x = 9x 2x + 4x = 6x -1x - 8x = -9x -2x - 4x = -6x

Check your answer by using FOIL

Always check for GCF before you do anything else. Lets do another example: Don’t Forget Method #1. Always check for GCF before you do anything else. Find a GCF Factor trinomial

When a>1 and c<1, there may be more combinations to try! Step 1:

Step 2: Order can make a difference!

O + I = 30 x - x = 29x This doesn’t work!! Step 3: Place the factors inside the parenthesis until O + I = bx. Try: F O I L This doesn’t work!! O + I = 30 x - x = 29x

Switch the order of the second terms and try again. F O I L This doesn’t work!! O + I = -6x + 5x = -x

Try another combination: Switch to 3x and 2x F O I L O+I = 15x - 2x = 13x IT WORKS!!

Factoring a perfect square trinomial in the form: Factoring Technique #3 continued Factoring a perfect square trinomial in the form:

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

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

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

Factoring Technique #4 Factoring By Grouping for polynomials with 4 or 3 terms

Factoring By Grouping 4 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 2: Factor out GCF from each group Example 1: Step 1: Group Step 2: Factor out GCF from each group Step 3: Factor out GCF again

Example 2:

Factoring Chart This chart will help you to determine which method of factoring to use. Type Number of Terms 1. GCF 2 or more 2. Diff. Of Squares 2 3. Trinomials 3

Here we go! 1) Factor y2 + 6y + 8 Use your factoring chart. Do we have a GCF? Is it a Diff. of Squares problem? Now we will learn Trinomials! You will set up a table with the following information. Nope! No way! 3 terms! Product of the first and last coefficients Middle coefficient The goal is to find two factors in the first column that add up to the middle term in the second column. We’ll work it out in the next few slides.

1) Factor y2 + 6y + 8 Create your MAMA table. Multiply Add +8 +6 Product of the first and last coefficients Middle coefficient Here’s your task… What numbers multiply to +8 and add to +6? If you cannot figure it out right away, write the combinations.

1) Factor y2 + 6y + 8 Place the factors in the table. Multiply Add +8 +6 +1, +8 -1, -8 +2, +4 -2, -4 +9, NO -9, NO +6, YES!! -6, NO Which has a sum of +6? We are going to use these numbers in the next step!

Now, group the first two terms and the last two terms. 1) Factor y2 + 6y + 8 Multiply Add +8 +6 +2, +4 +6, YES!! Hang with me now! Replace the middle number of the trinomial with our working numbers from the MAMA table y2 + 6y + 8 y2 + 2y + 4y + 8 Now, group the first two terms and the last two terms.

We have two groups! (y2 + 2y)(+4y + 8) Almost done! Find the GCF of each group and factor it out. y(y + 2) +4(y + 2) (y + 4)(y + 2) Tadaaa! There’s your answer…(y + 4)(y + 2) You can check it by multiplying. Piece of cake, huh? There is a shortcut for some problems too! (I’m not showing you that yet…) If things are done right, the parentheses should be the same. Factor out the GCF’s. Write them in their own group.

2) Factor x2 – 2x – 63 Create your MAMA table. Multiply Add -63 -2 Product of the first and last coefficients Middle coefficient -63, 1 -1, 63 -21, 3 -3, 21 -9, 7 -7, 9 -62 62 -18 18 -2 2 Signs need to be different since number is negative.

Replace the middle term with our working numbers. x2 – 2x – 63 x2 – 9x + 7x – 63 Group the terms. (x2 – 9x) (+ 7x – 63) Factor out the GCF x(x – 9) +7(x – 9) The parentheses are the same! Weeedoggie! (x + 7)(x – 9)

Here are some hints to help you choose your factors in the MAMA table. 1) When the last term is positive, the factors will have the same sign as the middle term. 2) When the last term is negative, the factors will have different signs.

2) Factor 5x2 - 17x + 14 Create your MAMA table. Multiply Add +70 -17 Product of the first and last coefficients Middle coefficient -1, -70 -2, -35 -7, -10 -71 -37 -17 Signs need to be the same as the middle sign since the product is positive. Replace the middle term. 5x2 – 7x – 10x + 14 Group the terms.

The parentheses are the same! Weeedoggie! (x – 2)(5x – 7) (5x2 – 7x) (– 10x + 14) Factor out the GCF x(5x – 7) -2(5x – 7) The parentheses are the same! Weeedoggie! (x – 2)(5x – 7) Hopefully, these will continue to get easier the more you do them.

Factor x2 + 3x + 2 (x + 2)(x + 1) (x – 2)(x + 1) (x + 2)(x – 1)

Factor 2x2 + 9x + 10 (2x + 10)(x + 1) (2x + 5)(x + 2) (2x + 2)(x + 5)

Factor 6y2 – 13y – 5 (6y2 – 15y)(+2y – 5) (2y – 1)(3y – 5)

Find the GCF! 2(x2 – 7x + 6) Now do the MAMA table! 2) Factor 2x2 - 14x + 12 Find the GCF! 2(x2 – 7x + 6) Now do the MAMA table! Multiply Add +6 -7 Signs need to be the same as the middle sign since the product is positive. -1, -6 -2, -3 -7 -5 Replace the middle term. 2[x2 – x – 6x + 6] Group the terms.

The parentheses are the same! Weeedoggie! 2[(x2 – x)(– 6x + 6)] Factor out the GCF 2[x(x – 1) -6(x – 1)] The parentheses are the same! Weeedoggie! 2(x – 6)(x – 1) Don’t forget to follow your factoring chart when doing these problems. Always look for a GCF first!!

Try These 1. a2 – 8a + 16 2. x2 + 10x + 25 3. 4y2 + 16y + 16 5. 3r2 – 18r + 27 6. 2a2 + 8a - 8

Video links Factoring Trinomials with Leading Coefficient of 1 Video: PH Factoring Trinomials of the Type x^2 + bx + c (where b is positive) Video: PH Factoring Trinomials of the Type x^2 - bx + c (where b is negative) Video: PH Factoring Trinomials of the Type x^2 + bx – c (where c is negative) Video: PH Factoring Perfect Square Trinomials

Video links Factoring Trinomials with Leading Coefficient not equal (≠) to 1 Video: PH Factoring Trinomials of the Type ax^2 + bx + c (where c is positive) Video: PH Factoring Trinomials of the Type ax^2 + bx - c (where c is negative) Video: PH Factoring perfect square trinomials with a ≠ 1

Video links Factoring by Grouping Factoring a four-term Polynomial by Grouping Factoring Trinomials by Grouping Video: PH Factoring Trinomials with Two Variables Video: PH Real World Factoring Problems

Try these on your own:

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