Algebra 2

Factoring Basics

&

Box Method

Factoring Polynomials

This process is basically the REVERSEof the distributive property.

)5)(2( xx 1032 xx

distributive property

factoring

1032 xx

In factoring you start with a polynomial (2 or more terms) and you want to rewrite it as a product (or as a single term)

Factoring Polynomials

Three terms

)5)(2( xx

One term

Techniques of Factoring Polynomials

1. Greatest Common Factor (GCF). The GCF for a polynomial is the largest monomial that divides each term of the polynomial.

Factor out the GCF:  23 24 yy

Factoring Polynomials - GCF

23 24 yy

y2

yyy22

Write the two terms in the form of prime factors…

They have in common 2yy

)12(2 2 yy

yy2

1)(2yy

This process is basically the reverse of the distributive property.

Check the work….

)12(2 2 yy 34y 22y

Factoring Polynomials - GCF

Factor the GCF:

24233 8124 cabcbaab

3 terms

4ab2( )b - 3a c2 + 2b c2 2

One term

Factoring Polynomials - GCF

)(

EXAMPLE:

)42(3)42(5 xxx

)42( x 5x - 3

Examples

Factor the following polynomial.

)53(4

)53(4

54432012

22

42

xx

xxxx

xxxxxxxx

Examples

Factor the following polynomial.

)15(3

)15(3

353315

42

42

42534253

xyyx

yxyx

yxyxyxyx

Techniques of Factoring Polynomials

2. Factoring a Polynomial with four or more Terms by Grouping

)2()3(

)3(2)3(

623

2

2

23

xx

xxx

xxx There is no GCF for allfour terms.

In this problem we factor GCFby grouping the first two terms and the last two terms.

To be continued….

3. Factoring Trinomials.

652 xx We need to find factors of 6

Since 6 can be written as the product of 2 and 3and 2 + 3 = 5, we can use the numbers 2 and 3 to factor the trinomial.

….that add up to 5

Techniques of Factoring Polynomials

Factoring Trinomials, continued...

652 xx 2 x 3 = 62 + 3 = 5

Use the numbers 2 and 3 to factor the trinomial…

Write the parenthesis, with An “x” in front of each.

3)2( xxWrite in the two numbers we found above.

xx )(

652 xxYou can check your work by multiplying back to get the original answer

3)2( xx

3)2( xx

6232 xxx

652 xx

So we factored the trinomial…

Factoring Trinomials, continued...

Factoring Trinomials

61

65

67

2

2

2

xx

xx

xx

Find factors of – 6 that add up to –5

Find factors of 6 that add up to 7

Find factors of – 6 that add up to 1

6 and 1

– 6 and 1

3 and –2

61

65

67

2

2

2

xx

xx

xx

factors of 6 that add up to 7: 6 and 1

1)6( xx

factors of – 6 that add up to – 5: – 6 and 1

factors of – 6 that add up to 1: 3 and – 2

1)6( xx

2)3( xx

Factoring Trinomials

Factoring TrinomialsThe hard case – “Box Method”

62 2 xx

Note: The coefficient of x2 is different from 1. In this case it is 2

62 2 xx

First: Multiply 2 and –6: 2 (– 6) = – 12

1

Next: Find factors of – 12 that add up to 1– 3 and 4

Factoring TrinomialsThe hard case – “Box Method”

62 2 xx

1. Draw a 2 by 2 grid.2. Write the first term in the upper left-hand corner 3. Write the last term in the lower right-hand corner.

22x6

Factoring TrinomialsThe hard case – “Box Method”

62 2 xx – 3 x 4 = – 12– 3 + 4 = 1

1. Take the two numbers –3 and 4, and put them, completewith signs and variables, in the diagonal corners, like this:

22x

6

It does not matter whichway you do the diagonal entries!

Find factors of – 12 that add up to 1

–3 x

4x

The hard case – “Box Method”1. Then factor like this:

22x6x3

x4

Factor Top Row Factor Bottom Row

2

22x6x3

x4x

From Left Column From Right Column

22x6x3

x42x

x222x

6x3

x4

x2

2x

3

x

The hard case – “Box Method”

22x6x3

x4

x2

2x

3

)32)(2(62 2 xxxx

Note: The signs for the bottom rowentry and the right column entry come from the closest term that youare factoring from. DO NOT FORGET THE SIGNS!!

++

Now that we have factored our box we can read offour answer:

The hard case – “Box Method”

24x12

x16x3

x

3

x44

12194 2 xx

Finally, you can check your work by multiplying back to get the original answer.

Look for factors of 48 that add up to –19 – 16 and – 3

)4)(34(12194 2 xxxx

Use “Box” method to factor the following trinomials.

1. 2x2 + 7x + 3

2. 4x2 – 8x – 21

3. 2x2 – x – 6

Check your answers.

1. 2x2 + 7x + 3 = (2x + 1)(x + 3)

2. 2x2 – x – 6 = (2x + 3)(x – 2)

3. 4x2 – 8x – 21 = (2x – 7)(2x + 3)

Note…

Not every quadratic expression can befactored into two factors.

• For example x2 – 7x + 13.

We may easily see that there are no factors of 13 that added up give us –7

• x2 – 7x + 13 is a prime trinomial.

Factoring the Difference of Two Squares

The difference of two bases being squared, factors as the product of the sum and difference of the bases that are being squared.

a2 – b2 = (a + b)(a – b) FORMULA:

(a + b)(a – b) = a2– ab + ab – b2 = a2 – b2

Factoring the difference of two squares

Factor x2 – 4y2 Factor 16r2 – 25

(x)2 (2y)2

(x – 2y)(x + 2y)

Now you can check the results…

(4r)2 (5)2

Difference of two squares

DifferenceOf two squares

(4r – 5)(4r + 5)

a2 – b2 = (a + b)(a – b)

Difference of two squares

)4)(4(

)4()(

16

22

2

yy

y

y

Difference of two squares

               

)95)(95(

)9()5(

8125

22

2

xx

x

x

Difference of two squares

                                                       

)4)(2)(2(

)4)(4(

)4()(

16

2

22

222

4

yyy

yy

y

y