5.4 factoring polynomials
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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
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