Factoring Quadratic Polynomials 1 copyright © 2011 Lynda Aguirre

Factoring Quadratic Polynomials

1copyright © 2011 Lynda Aguirre

Quadratic Polynomials

copyright (c) 2011 Lynda Aguirre 2

When two binomials are multiplied using the FOIL method,the answer can have 2, 3 or 4 terms.

)52)(34( yx 4 terms: Multiply this using FOIL

xxF 8)2)(4(: xyyxO 20)5)(4(:

6)2)(3(: I yyL 15)5)(3(:

yxyx 156208

This has no like terms, so we end up with a 4-term polynomial




)2)(34( xx3 terms: Multiply this using FOIL

24))(4(: xxxF xxO 8)2)(4(:

xxI 3))(3(: 6)2)(3(: L

6384 2 xxxThis has like terms, so add

them 654 2 xxThis polynomial now has 3-terms




)52)(52( xx2 terms: Multiply this using FOIL

24)2)(2(: xxxF xxO 10)5)(2(:

xxI 10)2)(5(: 25)5)(5(: L

2510104 2 xxxCombine like terms: they cancel out

254 2 x

This polynomial now has 2-terms


Quadratic PolynomialsWe just demonstrated that FOIL produces three types of polynomials

Four Terms Three Terms Two Terms

None of the terms combined

Middle terms combined Middle terms cancelled

yxyx 156208

654 2 xx 254 2 x

6384 2 xxx 2510104 2 xxx

A B C D A B C D A B C D

All four types of polynomials had 4-terms before cancelling or combining like terms

Because of this, we can use some form of the Because of this, we can use some form of the property AD=BC to see whether the property AD=BC to see whether the

polynomial is factorable for all three typespolynomial is factorable for all three types


yxyxx 151254 2 A B C D

A-D are the coefficients of each

of the four termsnot including the

signs

Plug in the numbers to see if both sides

are equal)12)(5()15)(4(

6060They’re equal which means this polynomial is factorable.

Check for Factorability: Four Terms Check for Factorability: Four Terms AD = BC AD = BC

AD=BC


1035414 yzyz

A B C D A-D are the

coefficients of eachof the four termsnot including the

signs Plug in the numbers to see if both sides

are equal

)35)(4()10)(14(

140140

They’re equal which means this polynomial is factorable.


AD=BC



aabb 62124 A B C D

A-D are the coefficients of each

of the four termsnot including the

signs

Plug in the numbers to see if both sides

are equal

)35)(4()10)(14(

140140They’re equal which means this polynomial is factorable.

AD=BC


Check for Factorability: Three Terms Check for Factorability: Three Terms AC method AC method

2255 2 xxx

First(A) Middle(B) Last(C) A B and C are the

coefficients of each term

Because the middle term came from combining the original middle terms, we have to alter our process to find them again.

10)2)(5(

235 2 xx

A B C D

Step 1: Multiply AC

Step 2: Find all factors of AC101

Step 3: Using the sign of the last term, add or subtract the factors of AC to see if

they equal the middle term

Sign of the last term: -

Subtract: 10-1 = 9

Subtract: 5-2 = 352

Step 4: These two factors were the original B and C

Middle term= 3


Check for Factorability: Three Terms Check for Factorability: Three Terms AC method AC method

3264 2 bbb

First(A) Middle(B) Last(C) A B and C are the

coefficients of each term

Because the middle term came from combining the original middle terms, we have to alter our process to find them again.

12)3)(4(

A B C D

Step 1: Multiply AC

Step 2: Find all factors of AC121

Step 3: Using the sign of the last term, add or subtract the factors of AC to see if

they equal the middle term

Sign of the last term: +

Add: 12+1 = 13Add: 6+2 = 862

Step 4: These two factors were the original B and C

Middle term= 8

384 2 bb

43 Add: 4+3 = 7


Check for Factorability: Two TermsCheck for Factorability: Two Terms

294 a

A D There are two terms because the B and C cancelled each other out. (i.e. they were the same number

with opposite signs)

Shortcut:

AD

Step 1 Calculate AD 36)9)(4( Step 2 Find factors of AD 361

1821239466

Step 3 Look for the repeating number

29664 aaa

ADCalculate :

636)9)(4(


Practice: Are these polynomials factorable?

239 2 xx

149 2 x

1324 2 xyyx

132 xx

142213 2 xxx

2564 2 a

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Factoring Quadratic Polynomials 1 copyright © 2011 Lynda Aguirre