Download 5.4 Factoring Polynomials

5.4 Factoring Polynomials
Group factoring
Special Cases
Simplify 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.
24 x  16 x  8 x  64
3
2
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.
24 x  16 x  8 x  64
3
2
Here it is 8


8 3x  2 x  x  8
3
2
Find the numbers that multiply to
the given number
The factors of 24 are
1 and 24
2 and 12
3 and 8
4 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 case
x2 + 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 case
x2 + 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 case
x2 + 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 Cubes
Different 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.
x  3x  4
2
x  7x  8
2
Quotients can be reduced by factoring
Simplify Quotients
Quotients can be reduced by factoring
x 2  3x  4 ( x  4)( x  1)

2
x  7 x  8 ( x  8)( x  1)
Common factors can be crossed out
Simplify Quotients
Common factors can be crossed out
x 2  3x  4 ( x  4)( x  1)

2
x  7 x  8 ( x  8)( x  1)
( x  4)
( x  8)
Simplify Quotients
Common factors can be crossed out
x 2  3x  4 ( x  4)( x  1)

2
x  7 x  8 ( x  8)( x  1)
( x  4)
( x  8)
Must stated that x cannot equal 1 or -8, why?
Simplify
Factor first
a a6
2
a  7a  10
2
Homework
Page 242 – 243
# 16 – 28 even; 29,
33, 35, 39, 47, 51
Must show work
Homework
Page 242 – 243
# 30, 31, 32, 34,
37, 38, 44, 46,
48, 50
Must show work