Algebra/Geometry Blend Unit #5: Factoring and Quadratic Functions Lesson 2: Factoring Trinomials. What does factoring really mean?

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1 Algebra/Geometry Blend Unit #5: Factoring and Quadratic Functions Lesson 2: Factoring Trinomials Name Period Date [page 1] Before you embark on your next factoring adventure, it is important to ask yourself this one, extremely important question: What does factoring really mean? Have you ever wanted to be on a game show? Now s your chance! You have been asked to be on Math Time a game show where all the questions are math related. Do you want to play? Let s do it! Follow the instructions below to begin. [Take a couple minutes and play the math game] In the game, you in the first round, then in the second round. Do you see how factoring relates to multiplying? Move to the next page to learn more about factoring and how it relates to polynomials. [page 2] Factoring Trinomials by Grouping There is a systematic approach to factoring trinomials with a leading coefficient greater than 1 called. If you need a refresher on factoring by grouping, select the water bottles. Click on

2 Take a moment to multiply these two binomials on your paper. (2x + 4)(x + 3) is the simplified product. Now, take an even closer look at the distributing and simplifying you just did. 2x 2 + 6x + 4x x x + 12 Notice, the two middle terms combine to give you the middle term of the simplified product. There is a special relationship among the coefficients of the simplified. This relationship occurs in every factorable trinomial! 2x 2 + 6x + 4x x x + 12 If you multiply the leading coefficient with the last term of the trinomial, you get 2 12 = 24. Now, think of two numbers that multiply to give you 24 and add to give you the coefficient of the middle term, 10. Did you come up with +6 and +4? 6 4 = = 10 Do those numbers look familiar? It's no coincidence that those two numbers are the middle terms of the product before simplifying! Check this out! [on the next page]

3 Let's say you had started with the product of 2x x + 12 and were asked to find the factors. Could you do the distribution method in reverse to come up with the factors? Absolutely! First, rewrite the middle term. Now, factor this four term polynomial by grouping! These are the two binomial factors that you started with! Let s work through another example and identify some steps that will guide you through the process of factoring trinomials by grouping. Factor completely: 3x x + 12 Step #1: Check for a GCF. 3x x + 12 There is no GCF. Move on to Step 2. Step #2: Split the middle term. Multiply the leading coefficient (3) and the last term of the trinomial (12). 3x x + 12 = 36 You need to find some combination of numbers that multiply to 36 but also add up to 13 Rewrite the polynomial with those factors replacing the middle term of the trinomial. 3x x x 2 + 9x + 4x + 12

4 Step #3: Factor by grouping. 3x 2 + 9x + 4x x 2 + 9x +4x x(x + 3) +4(x + 3) (x + 3)(3x + 4) Step #4: Check your factors. Multiply the binomials to make sure they bring you back to the original trinomial. (x + 3)(3x + 4) 3x 2 + 4x + 9x x x + 12 Done! Think about it! Does it matter which replace the middle term of the trinomial? you write the two terms in to No! It doesn t matter what order two terms are added in; the be the same. should still

5 Name Period Date HW: 5.02 Factoring Trinomials #1 Algebra/Geometry 1 Blend Factoring by Grouping Directions: Factor the following expression by grouping. Check your answers by multiplying the binomials. SHOW YOUR WORK! #1 x 3 2x 2 + 3x 6 #2 x 3 + 7x 2 + 3x + 21 #3 x 3 + 2x 2 8x 16 #4 x 3 3x 2 + 7x 21 #5 x 3 + 2x 2 3x 6 #6 x 3 5x 2 7x + 35

6 [page 3] Take a moment to watch this video (headphones!) [Follow along and write your work on the next page] 2x 2 3x 9 15x 2 27x 6

7 Take a look at the example below. Factor Completely: 8x 2 14x + 5 Step #1: Check for a Greatest Common Factor (GCF) No GCF. Step #2: Split the middle term. Multiply the leading coefficient (8) and the last term of the trinomial (5). 8 5 = 40 Find factors of this product (40) that add to give you the coefficient of the middle term ( 14). Rewrite the polynomial with those factors replacing the middle term of the trinomial. Step #3: Factor by grouping Separate the polynomial into two groups. Factor the GCF from the first group. The GCF of the first group is 4x. Factor the GCF from the last group. The GCF of the last group is 5. Factor the common binomial. Step #4: Check your factors using the distribution method. At the bottom of page 3, click on and show your work below. YOU SHOULD BE TRYING TO DO THESE, NOT JUST CLICKING ON CHECK YOUR ANSWER AND COPYING DOWN THE WORK. THAT MAKES NO SENSE. #1 3y 2 + 7y + 4

8 #2 4y 2 3y 1 #3 2x x + 12 #4 6x x + 9 #5 skip this one

9 Name Period Date HW: 5.02 Factoring Trinomials #2 Algebra/Geometry 1 Blend Factoring Trinomials Directions: Factor the following trinomials. Check your answers by multiplying the binomials. SHOW YOUR WORK! 1.) 2x x ) 3x x ) 5x x ) 4x x ) 6x x ) 8x x ) 12x x ) 12x x + 6

10 Name Period Date HW: 5.02 Factoring Trinomials #3 Algebra/Geometry 1 Blend Factoring Trinomials Directions: Factor the following trinomials. Check your answers by multiplying the binomials. SHOW YOUR WORK! 1.) 2x 2 + x 21 2.) 3x 2 10x 8 3.) 6x x ) 4x x 3 5.) 6x 2 23x ) 12x 2 5x 2 7.) 6x 2 + x 1 8.) 12x 2 8x 15

11 Factoring Trinomials x 2 + bx + c [page 4] Factor Completely: x 2 12x + 35 Step #1: Check for a GCF There is no GCF. Move onto Step 2. Step #2: Split the middle term Multiply the leading coefficient (1) and the last term of the trinomial (35) = 35 Find the factors of this product (35) that add to give you the coefficient of the middle term ( 12) Rewrite the polynomial with those factors replacing the middle term of the trinomial. Step #3: Factor by Grouping Step #4: Check your factors Did you notice that, when the leading coefficient is equal to 1, the factors of the last term of the trinomial that add up to the middle term are the same as the last terms in each binomial factor? This means you can skip the factor by grouping steps and write them directly into the binomials! Click on Example #2 and read through it. [Notice that when there is NOT a number in front of the x 2 term, factoring is WAY EASIER!] At the bottom of page 4, click on and show your work below. Try It Factor completely. Select Check Answer to check your work. #1 x 2 + 7x + 10 #2 x 2 10x + 24 #3 x 2 5x 24

12 Name Period Date HW: 5.02 Factoring Trinomials #4 Algebra/Geometry 1 Blend Factoring Trinomials Directions: Factor the following trinomials. Check your answers by multiplying the binomials. SHOW YOUR WORK! 1.) x 2 + 3x ) v 2 + 8v ) y y ) x 2 + 7x ) x x ) n n ) x 2 13x ) w 2 20w ) g 2 24g ) x x ) x 2 29x ) x x 38

13 Name Period Date HW: 5.02 Factoring Trinomials Day #5 Algebra/Geometry 1 Blend Factoring Trinomials Directions: Factor the following trinomials. Check your answers by multiplying the binomials. SHOW YOUR WORK! 1.) x 2 + 8x ) x x ) x 2 3x 10 4.) x 2 15x ) x x ) x 2 8x ) x 2 16x ) x x ) x 2 16x ) x x ) x 2 + 4x ) x 2 9x ) x 2 + 5x ) x x ) x 2 7x 30

14 [page 6] Click on Activity 1 and see how well you do! [page 7] Click on Activity 2 and see how well you do! Review It Factoring Trinomials of the Form ax 2 + bx + c Step 1: Check for a Step 2: Split the Multiply the coefficient and the term of the trinomial. Find factors of this product that to give you the coefficient of the middle term. Rewrite the polynomial with those factors replacing the middle term of the trinomial. Step 3: Factor by Step 4: Check your Factoring Trinomials of the Form x 2 + bx + c (when a=1) Step 1: Check for a Step 2: Write the x s as the term of each factor: (x )(x ). Step 3: Find the pair of numbers that to give you the last term, c, and to give you the coefficient of the middle term, b. Step 4: Fill in the term of each Step 5: Check your factors using the method. If a trinomial is not factorable, it is called.