main menu calculator tipsback to last sliderules factoring polynomials a presentation for the...

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Calculator TipsBack to last slide Rules

Factoring Polynomials

A presentation for the

greatest Algebra I kids at RJR

By Mrs. Sexton

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24142 xx

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RulesStep by Step

Easy Problems Medium Problems

Hard Problems Word Problems

23 93 xx 164 2 s

ppymmy 153204 22 2008 ts

656 2 yy

xxx 9156 23 322 8 x

Division of polynomial by monomial

Find dimensions when area is given

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Rules for Factoring Polynomials

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Step by Step

• Is there a GCF?– Yes

• Factor as the product of the GCF and one other factor—i.e. GCF•(the other factor). Look at the other factor and go to the next step below with it.

– No• Go the the next step.

• Is it a binomial?– Yes

• Is it a difference of two squares? (a2-b2)– Yes—Factor as (a+b)(a-b).– No—It can’t be factored any more.

– No• Go to the next step.

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• Is it a trinomial?– Yes

• Do you recognize it as a pattern for a perfect square trinomial? (a2+2ab+b2) or (a2-2ab+b2)

– Yes—Factor as (a+b)2 or (a-b)2 – No—Go to next step.

• Use the ac and b pattern to look for factors.• Can you find factors of ac that add up to b?

– Yes—Rewrite the equation with those factors, group, and factor.

– No—You can’t do anything else. If there’s no GCF, it’s a prime polynomial.

– No• Go to the next step.

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• Is it a four-term polynomial?– Yes

• Are there two sets of terms that you can group together that have a common factor?

– Yes—Group and factor.

– No—If it doesn’t have a GCF, it’s a prime polynomial.

– No• If it doesn’t have a GCF, it’s a prime polynomial.

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NOTE:

At EVERY step along the way, you must look at the factors that you get to see if they can be factored any more.

Factoring completely means that no factors can be broken down any further using any of the rules you’ve learned.

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Practice

24142 xx

Factor completely.

Is there a GCF?

No. Is it a binomial, trinomial, or four-term polynomial?

It’s a trinomial.

Do you recognize it as a perfect square trinomial?

No. Use ac and b.

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Use your handy-dandy calculator or your super math skills to find 12 and 2 as the factors to use.

ac b

1 • 24 14

24

12, 2

Rewrite the equation with those two factors in the middle.

242122 xxx

24142 xxGroup.

)242()12( 2 xxx Factor out the GCF from each group.

)12(2)12( xxx Write the two factors.

)2)(12( xx Neither one of these factors can be broken down any more, so you’re done.

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23 93 xx Factor completely

Is there a GCF?

Is it a binomial, trinomial, or four-term polynomial?

Yes. Write the GCF first and the remaining factor after it.

)3(3 2 xx Look at the remaining factor. (x-3)

It’s a binomial. Is it a difference of two squares? (a2-b2)

No. You can’t do anything else.

)3(3 2 xx is the completely factored form.

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Factor completely

164 2 s Is there a GCF?


)4(4 2 s Look at the remaining factor. (s2-4)



Yes. s2 is a square (s • s) and 4 is a square (2 • 2). Factor as (s+2)(s-2). Then write the complete factorization.

)2)(2(4 ss

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Factor completely

ppymmy 153204 Is there a GCF?

No. There is no single factor that goes into all four of the terms.


It’s a four-term polynomial. Factor by grouping.

)153()204( ppymmy Factor out the GCF from each group.)5(3)5(4 ypym

)5)(34( ypm Write the two factors.

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Factor completely

)25(8 22 ts

Is there a GCF?


Look at the remaining factor. (s2-25t2)



Yes. s2 is a square (s • s) and 25t2 is a square (5t • 5t). Factor as (s+5t)(s-5t). Then write the complete factorization.

)5)(5(8 tsts

22 2008 ts

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Factor completely

656 2 yy Is there a GCF?

Is it a binomial, trinomial, or four-term polynomial?No.

It’s a trinomial.


No. Use ac and b.

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ac b

6 • -6 -5

-36

4, -9

Look for factors of –36 that add up to –5. Use your calculator or your math skills to find 4 and -9 as the factors to use.


)64()96( 2 yyy

656 2 yyGroup.6496 2 yyyFactor out the GCF from each group.

)32(2)32(3 yyy Write the two factors.

)23)(32( yy

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xxx 9156 23 Factor completely

Is there a GCF?


)352(3 2 xxx Look at the remaining factor.

)352( 2 xx


It’s a trinomial.


No. Use ac and b.

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ac b

2 • -3 5

-6

6, -1

Look for factors of -6 that add up to 5. Use your calculator or your math skills to find 6 and -1 as the factors to use.


Group. Remember to change the –3 to a +3 because of the minus sign in the grouping!!

Factor out the GCF from each group.Write all three factors.

)352(3 2 xxx

]3162[3 2 xxxx

)]31()62[(3 2 xxxx)]3(1)3(2[3 xxxx

)12)(3(3 xxx

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322 8 x

Factor completely

Is there a GCF?


)16(2 8 x Look at the remaining factor.

)16( 8 x


Yes. x8 is a square (x4 • x4) and 16 is a square (4 • 4). Factor as (x4 + 4)(x4 - 4).


So far we have 2(x4 + 4)(x4 - 4). (Please continue—not done yet!!)

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2(x4 +4)(x4 -4)Look at what you have. Can either of the binomials be broken down?

(x4 +4)

Is this binomial a difference of two squares? (a2-b2)

No. It can’t be broken down. So, we have to keep this factor.

(x4 -4)Is this binomial a difference of two squares? (a2-b2)

Yes. x4 is a square (x2 • x2) and 4 is a square (2 • 2). Factor as (x2 + 2)(x2 - 2).

What about the other binomial?

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Put it all together.

322 8 x

)16(2 8 x

2(x4 +4)(x4 -4)

2(x4 +4)(x2 +2)(x2 -2)

Not a difference of squares. Can’t go any farther!!

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Word Problem #1

What is the quotient when

xxx 16812 23 is divided by 4x?

23x

This question is asking you to find the OTHER FACTOR after you take out the greatest common factor of 4x.

Simplify each term.

x

x

x

x

x

x

x

xxx

4

16

4

8

4

12

4

16812 2323

x2 4

423 2 xx

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Word Problem #2

A rectangular garden plot has an area represented by the expression

28318 2 xx

Find the dimensions of the garden plot.

This is a factoring problem. You need to find the two factors that multiply together to give you 28318 2 xx

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Is there a GCF?

Is it a binomial, trinomial, or four-term polynomial?No.

It’s a trinomial.


No. Use ac and b.

28318 2 xx

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ac b

18 • -28 -3

-504

21, -24

Look for factors of –504 that add up to –3. Use your calculator or your math skills to find 21 and -24 as the factors to use.


)2821()2418( 2 xxx

Group.28212418 2 xxxFactor out the GCF from each group.

)43(7)43(6 xxx Write the two factors.

)76)(43( xx

28318 2 xx

Length is 3x - 4 and width is 6x + 7

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Calculator Tips

To find factors of the ac term, use the following steps in your calculator:

•Press the Y= button.

•In Y1=, type the ac value / X.

•In Y2=, type X + VARS, arrow to Y_VARS, Enter, Enter

•Go to Table and look for the b in column Y2. When you find it, use the values in the X column and the Y1 column as your two factors to put in the equation. IF YOU CAN’T find the b value in the Y2 column, the trinomial can’t be factored.

•NOTE: Remember that you might need to scroll up the screen to find negative numbers that give you the correct value in the Y2 column.

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