5.4 Factoring Polynomials5.4 Factoring Polynomials

Group factoring Group factoring

Special CasesSpecial Cases

Simplify QuotientsSimplify Quotients

The first thing to do when factoring

Find the greatest common factor (G.C.F), this is a number that can divide into all the terms of the polynomial.

6481624 23 xxx

The first thing to do when factoring

Find the greatest common factor (G.C.F), this is a number that can divide into all the terms of the polynomial.

Here it is 8

6481624 23 xxx

8238 23 xxx

Find the numbers that multiply to the given number

The factors of 24 are 1 and 242 and 123 and 84 and 6-1 and -24-2 and -12-3 and -8-4 and -6

There are only one set of numbers that are factors of 24 and add to 11

So in the trinomial x2 + 11x + 24

(x + ___)(x + ___)

Since we add the like terms of the inner and outer parts of FOIL and multiply them to be the last number; the only numbers would work would be 3 and 8.

Factor

x2 _ 2x - 63

Group factoring

Here you will factor four terms, two at a time.

When you find the common binomial that they have in common, factor it out of both parts.

x3 + 5x2 – 2x – 10

x2 (x + 5) – 2(x + 5)

(x + 5)(x2 -2)

Group factoring

Here you will factor four terms, two at a time.

When you find the common binomial that they have in common, factor it out of both parts.

x3 + 5x2 – 2x – 10

x2 (x + 5) – 2(x + 5)

(x + 5)(x2 -2)

Group factoring

Here you will factor four terms, two at a time.

When you find the common binomial that they have in common, factor it out of both parts.

x3 + 5x2 – 2x – 10

x2 (x + 5) – 2(x + 5)

(x + 5)(x2 -2)

Factor x3 + 4x2 + 2x + 8

x2 ( x + 4) + 2(x + 4)

(x + 4)(x2 + 2)

Factor x3 + 4x2 + 2x + 8

x2 ( x + 4) + 2(x + 4)

(x + 4)(x2 + 2)

Factor x3 + 4x2 + 2x + 8

x2 ( x + 4) + 2(x + 4)

(x + 4)(x2 + 2)

How would you factor 3y2 – 2y -5

I would turn it into a group factoring problem

How would you factor 3y2 – 2y -5

I would turn it into a group factoring problem

Multiply the end together, 3 times – 5 is -15.

What multiplies to be -15 and adds to – 2

- 5 and 3

So I break the middle term into -5y and +3y

3y2 - 5y + 3y – 5

How would you factor 3y2 – 2y -5

3y2 - 5y + 3y – 5

y(3y – 5) + 1(3y – 5)

(3y – 5)(y + 1)

How would you factor 3y2 – 2y -5

3y2 - 5y + 3y – 5

y(3y – 5) + 1(3y – 5)

(3y – 5)(y + 1)

How would you factor 3y2 – 2y -5

3y2 - 5y + 3y – 5

y(3y – 5) + 1(3y – 5)

(3y – 5)(y + 1)

Homework

Page 242

# 4 – 8

15 – 27 odd

Must show work

Special Case

Factor x2 – 25

Here you find with multiply to be -25 and adds to be 0. The number must be negative and positive.

( x + __)(x - __)

The rule is a2 – b2 = (a +b)(a – b)

16x2 – 81y4

(4x)2 – (9y2)2 = (4x + 9y2)(4x – 9y2)

Another Special casex2 + 2xy + y2 = (x + y)2

x2 - 2xy + y2 = (x - y)2

4x2 – 20xy + 25y2

(2x)2 – (2x)(5y) + (5y)2

(2x – 5y)2

Another Special casex2 + 2xy + y2 = (x + y)2

x2 - 2xy + y2 = (x - y)2

4x2 – 20xy + 25y2

(2x)2 – (2x)(5y) + (5y)2

(2x – 5y)2

Another Special casex2 + 2xy + y2 = (x + y)2

x2 - 2xy + y2 = (x - y)2

4x2 – 20xy + 25y2

(2x)2 – (2x)(5y) + (5y)2

(2x – 5y)2

Sum of two CubesDifferent of Cubes

a3 + b3 = (a + b)(a2 – ab + b2)

x3 + 343y3 = (x + 7y)(x2 – x(7y) + (7y)2)

x3 + 343y3 = (x + 7y)(x2 – 7xy + 49y2)

a3 - b3 = (a - b)(a2 + ab + b2)

8k3 – 64c3 = (2k – 4c)((2k)2 + (2k)(4c) + (4c)2)

8k3 – 64c3 = (2k – 4c)(4k2 + 8ck + 16c2)

Factor: x3y3 + 8

Sum of Cubes

(xy)3 + (2)3

((xy) + (2))((xy)2 – (xy)(2) + (2)2)

(xy + 2)(x2y2 – 2xy +4)

Factor: x3y3 + 8

Sum of Cubes

(xy)3 + (2)3

((xy) + (2))((xy)2 – (xy)(2) + (2)2)

(xy + 2)(x2y2 – 2xy +4)

Factor: x3y3 + 8

Sum of Cubes

(xy)3 + (2)3

((xy) + (2))((xy)2 – (xy)(2) + (2)2)

(xy + 2)(x2y2 – 2xy +4)

Factor: x3y3 + 8

Sum of Cubes

(xy)3 + (2)3

((xy) + (2))((xy)2 – (xy)(2) + (2)2)

(xy + 2)(x2y2 – 2xy +4)

Simplify Quotients

Quotients are fractions with variables.

Quotients can be reduced by factoring

87

432

2

xx

xx

Simplify Quotients

Quotients can be reduced by factoring

Common factors can be crossed out

)1)(8(

)1)(4(

87

432

2

xx

xx

xx

xx

Simplify Quotients

Common factors can be crossed out

)8(

)4(

)1)(8(

)1)(4(

87

432

2

x

x

xx

xx

xx

xx

Simplify Quotients

Common factors can be crossed out

Must stated that x cannot equal 1 or -8, why?

)8(

)4(

)1)(8(

)1)(4(

87

432

2

x

x

xx

xx

xx

xx

Simplify

Factor first

107

62

2

aa

aa

HomeworkHomework

Page 242 – 243 Page 242 – 243

## 16 – 28 even; 29, 16 – 28 even; 29,

33, 35, 39, 47, 5133, 35, 39, 47, 51

Must show workMust show work

HomeworkHomework

Page 242 – 243 Page 242 – 243

## 30, 31, 32, 34, 30, 31, 32, 34,

37, 38, 44, 46, 37, 38, 44, 46,

48, 5048, 50

Must show workMust show work