Factoring Trinomials By Grouping

Algebra/Geometry BlendUnit #5: Factoring and Quadratic FunctionsLesson 2: Factoring TrinomialsNamePeriodDate[page 1]Before you embark on your next factoring adventure, it isimportant to ask yourself this one, extremely importantquestion:What does factoring really mean?Have you ever wanted to be on a game show? Now’s yourchance! You have been asked to be on Math Time—a gameshow where all the questions are math related. Do you want toplay? Let’s do it! Follow the instructions below to begin.[Take a couple minutes and play the math game]In the game, youin the first round, thenin 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 GroupingThere is a systematic approach to factoring trinomials with a leading coefficient greaterthan 1 called.If you need a refresher on factoring by grouping, select the water bottles.Click on

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.2x2 6x 4x 122x2 10x 12Notice, the two middle terms combine to give you the middle term of the simplifiedproduct. There is a special relationship among the coefficients of the simplified. Thisrelationship occurs in every factorable trinomial!2x2 6x 4x 122x2 10x 12If 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 thecoefficient of the middle term, 10.Did you come up with 6 and 4?6 4 246 4 10Do those numbers look familiar? It's no coincidence that those two numbers are themiddle terms of the product before simplifying!Check this out! [on the next page]

Let's say you had started with the product of 2x2 10x 12 and were asked to find thefactors. 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 throughthe process of factoring trinomials by grouping.Factor completely: 3x2 13x 12Step #1: Check for a GCF.3x2 13x 12There 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).3x2 13x 12 36You 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.3x2 13x 123x2 9x 4x 12

Step #3: Factor by grouping.3x2 9x 4x 123x2 9x 4x 123x(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)3x2 4x 9x 123x2 13x 12Done!Think about it! Does it matter whichreplace the middle term of the trinomial?you write the two terms in toNo! It doesn’t matter what order two terms are added in; thebe the same.should still

NameHW: 5.02 –Factoring Trinomials #1PeriodDateAlgebra/Geometry 1 BlendFactoring by GroupingDirections: Factor the following expression by grouping. Check your answers bymultiplying the binomials. SHOW YOUR WORK!#1x3 – 2x2 3x – 6#2x3 7x2 3x 21#3x3 2x2 – 8x – 16#4x3 – 3x2 7x – 21#5x3 2x2 – 3x – 6#6x3 – 5x2 – 7x 35

[page 3]Take a moment to watch this video (headphones!)[Follow along and write your work on the next page]2x2 – 3x – 915x2 – 27x – 6

Take a look at the example below.Factor Completely:8x2 – 14x 5Step #1: Check for a Greatest Common Factor (GCF) No GCF.Step #2: Split the middle term. Multiply the leading coefficient (8) and the last termof the trinomial (5).8 5 40 Find factors of this product (40) that add to give youthe coefficient of the middle term ( 14).Rewrite the polynomial with those factors replacingthe 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 thefirst group is 4x. Factor the GCF from the last group. The GCF of thelast group is 5. Factor the common binomial.Step #4: Check your factors using the distributionmethod.At the bottom of page 3, click onand show your work below.YOU SHOULD BE TRYING TO DO THESE, NOT JUST CLICKING ON CHECK YOUR ANSWERAND COPYING DOWN THE WORK. THAT MAKES NO SENSE.#13y2 7y 4

#24y2 – 3y – 1#32x2 11x 12#46x2 15x 9#5skip this one

NameHW: 5.02 –Factoring Trinomials #2PeriodDateAlgebra/Geometry 1 BlendFactoring TrinomialsDirections: Factor the following trinomials. Check your answers by multiplying thebinomials. SHOW YOUR WORK!1.)2x2 13x 152.)3x2 13x 43.)5x2 17x 64.)4x2 27x 185.)6x2 11x 36.)8x2 14x 37.)12x2 23x 108.)12x2 17x 6

NameHW: 5.02 –Factoring Trinomials #3PeriodDateAlgebra/Geometry 1 BlendFactoring TrinomialsDirections: Factor the following trinomials. Check your answers by multiplying thebinomials. SHOW YOUR WORK!1.)2x2 x – 212.)3x2 – 10x – 83.)6x2 19x 104.)4x2 11x – 35.)6x2 – 23x 206.)12x2 – 5x – 27.)6x2 x – 18.)12x2 – 8x – 15

[page 4]Factoring Trinomials x2 bx cFactor Completely:x2 – 12x 35Step #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 lastterm of the trinomial (35).1 35 35 Find the factors of this product (35) that add to giveyou the coefficient of the middle term ( 12) Rewrite the polynomial with those factorsreplacing the middle term of the trinomial.Step #3: Factor by GroupingStep #4: Check your factorsDid you notice that, when the leading coefficient is equal to 1, the factors of the last termof the trinomial that add up to the middle term are the same as the last terms in eachbinomial factor?This means you can skip the factor by grouping stepsand write them directly into the binomials!Click on Example #2 and read through it. [Notice that when there is NOT a number in frontof the x2 term, factoring is WAY EASIER!]At the bottom of page 4, click onand show your work below.Try ItFactor completely. Select Check Answer to check your work.#1x2 7x 10#2x2 10x 24#3x2 5x 24

NameHW: 5.02 –Factoring Trinomials #4PeriodDateAlgebra/Geometry 1 BlendFactoring TrinomialsDirections: Factor the following trinomials. Check your answers by multiplying thebinomials. SHOW YOUR WORK!1.)x2 3x 27.)v2 8v 152.)y2 14y 248.)x2 7x 123.)x2 13x 309.)n2 11n 104.)x2 – 13x 4210)w2 – 20w 1005.)g2 – 24g 14411.) x2 50x 4006.)x2 – 29x 10012.) x2 17x – 38

NameHW: 5.02 –Factoring Trinomials Day #5PeriodDateAlgebra/Geometry 1 BlendFactoring TrinomialsDirections: Factor the following trinomials. Check your answers by multiplying thebinomials. SHOW YOUR WORK!1.)x2 8x 152.)x2 10x 163.)x2 – 3x – 104.)x2 – 15x 565.)x2 10x 216.)x2 – 8x 167.)x2 – 16x 608.)x2 10x 249.)x2 – 16x 6410.) x2 10x – 2411.) x2 4x – 6012.) x2 – 9x 2013.) x2 5x – 2414.) x2 20x 10015.) x2 – 7x – 30

[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 ItFactoring Trinomials of the Form ax2 bx cStep 1: Check for aStep 2: Split the Multiply thecoefficient and theterm of the trinomial. Find factors of this product thatto give you the coefficient of the middleterm. Rewrite the polynomial with those factors replacing the middle term of thetrinomial.Step 3: Factor byStep 4: Check yourFactoring Trinomials of the Form x2 bx c (when a 1)Step 1: Check for aStep 2: Write the x’s as theterm of eachfactor: (x )(x ).Step 3: Find the pair of numbers thatto give you the last term, c,andto give you the coefficient of the middle term, b.Step 4: Fill in theterm of eachStep 5: Check your factors using theIf a trinomial is not factorable, it is calledmethod.

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