Factoring Polynomials Grouping, Trinomials, Binomials, GCF & Solving Equations Adapted by Christopher Carnes, Rancho Viejo MS, Hemet, CA

Essential Question: How can we use various techniques to factor a polynomial?

Factoring Four Term Polynomials

Factor by Grouping When polynomials contain four terms, it is sometimes easier to group like terms in order to factor. Your goal is to create a common factor. You can also move terms around in the polynomial to create a common factor. Practice makes you better in recognizing common factors.

Factor by Grouping Example 1: FACTOR: 3xy - 21y + 5x – 35 Factor the first two terms: 3xy - 21y = 3y (x – 7) Factor the last two terms: + 5x - 35 = 5 (x – 7) The green parentheses are the same so it’s the common factor Now you have a common factor (x - 7) (3y + 5)

Factor by Grouping Example 2: FACTOR: 6mx – 4m + 3rx – 2r Factor the first two terms: 6mx – 4m = 2m (3x - 2) Factor the last two terms: + 3rx – 2r = r (3x - 2) The green parentheses are the same so it’s the common factor Now you have a common factor (3x - 2) (2m + r)

Factor by Grouping Example 3: FACTOR: 15x – 3xy + 4y –20 Factor the first two terms: 15x – 3xy = 3x (5 – y) Factor the last two terms: + 4y –20 = 4 (y – 5) The green parentheses are opposites so change the sign on the 4 - 4 (-y + 5) or – 4 (5 - y) Now you have a common factor (5 – y) (3x – 4)

Factoring Trinomials

Factoring Trinominals When trinomials have a degree of “2”, they are known as quadratics. We learned earlier to use the “diamond” to factor trinomials that had a “1” in front of the squared term. x2 + 12x + 35 (x + 7)(x + 5)

More Factoring Trinomials When there is a coefficient larger than “1” in front of the squared term, we can use a modified diamond or square to find the factors. Always remember to look for a GCF before you do ANY other factoring.

More Factoring Trinomials Let’s try this example 3x2 + 13x + 4 Make a box Write the factors of the first term. Write the factors of the last term. Multiply on the diagonal and add to see if you get the middle term of the trinomial. If so, you’re done!

Difference of Squares

Difference of Squares When factoring using a difference of squares, look for the following three things: only 2 terms minus sign between them both terms must be perfect squares If all 3 of the above are true, write two ( ), one with a + sign and one with a – sign : ( + ) ( - ).

Try These 1. a2 – 16 2. x2 – 25 3. 4y2 – 16 4. 9y2 – 25 5. 3r2 – 81 6. 2a2 + 16

Perfect Square Trinomials

Perfect Square Trinomials When factoring using perfect square trinomials, look for the following three things: 3 terms last term must be positive first and last terms must be perfect squares If all three of the above are true, write one ( )2 using the sign of the middle term.

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

Factoring Completely

Factoring Completely Now that we’ve learned all the types of factoring, we need to remember to use them all. Whenever it says to factor, you must break down the expression into the smallest possible factors. Let’s review all the ways to factor.

Types of Factoring Look for GCF first. Count the number of terms: 4 terms – factor by grouping 3 terms - look for perfect square trinomial if not, try diamond or box 2 terms - look for difference of squares If any ( ) still has an exponent of 2 or more, see if you can factor again.

Solving Equations by Factoring

Steps to Solve Equations by Factoring We know that an equation must be solved for the unknown. Up to now, we have only solved equations with a degree of 1. 2x + 8 = 4x +6 -2x + 8 = 6 -2x = -2 x = 1

Steps to Solve Equations by Factoring If an equation has a degree of 2 or higher, we cannot solve it until it has been factored. You must first get “0” on one side of the = sign before you try any factoring. Once you have “0” on one side, use all your rules for factoring to make 2 ( ) or factors.

Steps to Solve Equations by Factoring Next, set each factor = 0 and solve for the unknown. x2 + 12x = 0 Factor GCF x(x + 12) = 0 (set each factor = 0, & solve) x = 0 x + 12 = 0 x = - 12 You now have 2 answers, x = 0 and x = -12.