FACTORINGBINOMIALS

Factoring Binomials(4 Possibilities)

• GCF (Greatest Common Factor)

• Difference of Squares

•Sum or Difference of Cubes

• Prime (Not Factorable)

If a GCF exists, simply factor it out.

10y5 −12y=

3x2 −18x=

5x2 −20x5 =

2y(5y 4−6)

3x(x−6)

5x2 (1−4x3)

2nd PossibilityDIFFERENCE OF 2

SQUARES

If both terms in the binomial are squares and they are subtracted, a Simple Formula will give you the

answer.

a2 −b2( ) = a+b( ) a−b( )

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x 2 − 25 = (x + 5)(x − 5)

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x 2 − 25 = ?

a2 −b2 =(a+b)(a−b)

a = x2 =x

Example 1)

The Difference of Squares Formula is

Find a and b, then plug them into the formula.

b = 25 =5

Examples of Diff. Of Squares

y2 −49 =

x2 −25 =

x4 −4 =

9x2 −y2 =

A2 −B2 = (A + B)(A−B)

(x + 5)(x−5)

(x2 + 2)(x2 −2)

(3x + y)(3x−y)(y + 7)(y−7)

Recognizing Monomial Squares

Numbers that are perfect squares are: 1, 4, 9, 16, 25, 36…

Variables that are perfect squares are:(Any even powered variable is a perfect square)

Monomials that are perfect squares are:€

x 2,y 2,x 4,x10,y16

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36x 2,4y 2,25x 4 ,9x10,100y16

2x2

2x2 4x4

Check this picture out.It shows why any even poweredVariable is a perfect square.

Try These

€

y 2 −1 = ?

€

x 2 −100 = ?

€

x 4 −144 = ?

€

16x 2 − 4y 2 = ?

3nd PossibilitySUM or DIFFERENCE OF 2 CUBES

If both binomials are cubes and they are added

( )( )2233 babababa +−+=+

If both binomials are cubes and they are subtracted

( )( )2233 babababa ++−=−

To solve sum and difference of two cubes, simply solve for a and b. Then plug into the correct formula.

( )( )2233 babababa +−+=+

( )( )2233 babababa ++−=−

1. Factor completely 12527 3 +x

12527 3 +x ( ) ( ) ( )( )2233 )5()5)(3()3(5353 +−+=+ xxxx

( )( )25159(53 2 +−+= xxx

2. Factor completely xx 4487 4 −

( )647 3 −= xx

( )( )16447 2 ++−= xxxx

GCF FIRST!

a b

a = xb = 4

a = 3xb = 5

If the binomial does not have a GCF & is not a Diff. Of Squares , Diff. of Cubes, or Sum of Cubes

PRIME & NOT FACTORABLE

4th and last possibility when trying to factor a binomial

Examples of Prime Binomials

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x 2 − 2

€

4y 2 + 25

€

4x 2 − y 3

Which binomials are Prime?

€

33x 2 − 9

€

49x 2 − y

€

121x 2 −100€

4x 2 − y10