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Rules of Factoring Polynomials

A presentation for RHHS Grade 10 Wonderful Students

By Ms. Wang

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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 of Factoring

Main MenuFlowchart of Factoring polynomials

GCF and Leading “-”Factor out GCF and rewrite the left polynomial

inside a parenthesis

Binomial Trinomial

Difference of two squares XBOX

22 ba

))(( baba ))(( dcxbax

dcx

bax

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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.

NoGo 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.

NoGo to the next step.

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Is it a trinomial?Yes

Use the X BOX pattern to look for factors.

NoGo to the next step.

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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 or trinomial?

It’s a trinomial.

XBOX

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

Rewrite the equation with those two factors in the middle.

24142 xx

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

Neither one of these factors can be broken down any more, so you’re done.

24

x

x

12

2

x2

12x + 2x = 14x

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23 93 xx Factor completely

Is there a GCF?

Is it a binomial or trinomial?

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?

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

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

Is it a binomial, trinomial?

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

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

656 2 yy Is there a GCF?

Is it a binomial or trinomial?No.

It’s a trinomial.

Main Menu6y2 b

2y -3

3y 2

Check the sum of the cross product is -5y

Rewrite the equation with those two factors in the middle.

656 2 yy

Write the two factors.)23)(32( yy

2*2y + 3y *(-3) = -5y

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xxx 9156 23 Factor completely

Is there a GCF?

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

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

)352( 2 xx

Is it a binomial or trinomial?

It’s a trinomial.

Main Menu2x2 -3

2x -1

x 3

Check the sum of the cross product is 5x.

Rewrite the equation with those two factors in the middle.

Write all three factors.

)352(3 2 xxx

)12)(3(3 xxx

2x*3 + x*(-1) =5x

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322 8 x

Factor completely

Is there a GCF?

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

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

)16( 8 x

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

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

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

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.

28318 2 xx

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3x -4

6x 7

Check the sum of the cross product of the XBOX is -3x.

Rewrite the equation with those two factors in the middle.

Write the two factors.)76)(43( xx

28318 2 xx

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

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The End

Practice Makes Master!