5.4 Factoring Polynomials
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Transcript of 5.4 Factoring Polynomials
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
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