Download x - GCC

Chapter 5
MAT 090/92
Section 5.1 Greatest Common Factor & Factoring by Grouping.
Factors: Numbers, variables and/or quantities that are multiplied together.
Example: Find all possible ways to factor 12x3
Greatest Common Factor (G.C.F.) is the largest factor that is common to every
term.
Examples: What is the GCF of the following…
12 and 18
24 and 32
Factor the following polynomials by the GCF.
12 x − 18
24a + 32b
16, 24, and 36
16 x 2 + 24 x − 36
What property are we doing backwards?_____________________________
Factor the following polynomials by the GCF.
12 x + 3
4a 2b + 40ab 2
− 3 x 5 y 2 − 15 x 3 y 3 − 6 x 3 y 5 + 9 x 2 y 4
6 x 5 − 24 x 3 − 18 x 2
2a( x + y ) − 3b( x + y )
Chapter 5
MAT 090/92
Factor by grouping is where there is not a GCF in all the terms, but there are
GCF’s in groups of terms.
Step 1.
5 x 3 − x 2 + 15 x − 3
Step 2.
2 x3 + 8x 2 + x + 4
8 x 4 + 6 x − 28 x 3 − 21
6ax + 3bx + 4ay + 2by
6 w + 10 wx − 35 x − 21
Chapter 5
MAT 090/92
2
Section 5.2 Factoring Trinomials in the form x + bx + c .
Factoring trinomials is working the _________ process backwards.
Sign pattern in FOILing, Binomials to trinomial.
F O
I
L
(x + 2)(x + 5) =
x 2 + 5 x + 2 x + 10 =
x 2 + 7 x + 10
(x − 2)(x − 5) =
x 2 − 5 x − 2 x + 10 =
x 2 − 7 x + 10
(x + 2)(x − 5) =
x 2 − 5 x + 2 x − 10 =
x 2 − 3x − 10
(x − 2)(x + 5) =
x 2 + 5 x − 2 x − 10 =
x 2 + 3x − 10
There is another pattern. Find the product of the F & L terms and O & I terms.
Make up any two binomials and FOIL them.
This pattern gives us the a, b, c rule for finding our factors.
Number Sense!
Chapter 5
MAT 090/92
Factor the trinomials.
x 2 − 8 x + 12
x 2 − 8 x − 20
x 2 − 5x + 6
x 2 + 5x − 6
x 2 − 4 x − 21
x 2 + 10 x + 24
x 2 − 10 x − 24
x 2 − 5 x − 24
x 2 − 11x + 24
x 2 − 20 x + 75
x 2 − 8 x − 48
x 2 − 7 x + 60
Chapter 5
MAT 090/92
Factor the trinomials. If the trinomial can not be factored, then call it prime.
x 2 − 20 x + 36
x 2 − x − 20
x2 − 7x + 6
x2 − 7x + 8
x 2 + 12 xy − 45 y 2
m 2 − 5mn − 24n 2
x 4 + x 3 − 132x 2
− 3x 8 + 12 x 7 + 15 x 6
Factor completely.
5 x 2 − 5 x − 30
− 2 x 2 − 12 x − 36
4 x 3 − 8 x 2 y 2 − 48 y 3
Chapter 5
MAT 090/92
2
Section 5.3 Factoring Trinomials in the form ax + bx + c .
Factoring trinomials is working the _________ process backwards.
Sign pattern in FOILing, Binomials to trinomial.
F
O
I
L
(3x + 2)(2 x + 5) =
6 x 2 + 15 x + 4 x + 10 =
6 x 2 + 19 x + 10
(3x − 2)(2 x − 5) =
6 x 2 − 15 x − 4 x + 10 =
6 x 2 − 19 x + 10
(3x + 2)(2 x − 5) =
6 x 2 − 15 x + 4 x − 10 =
6 x 2 − 11x − 10
(3x − 2)(2 x + 5) =
6 x 2 + 15 x − 4 x − 10 =
6 x 2 + 11x − 10
Find the product of the F & L terms and O & I terms.
Chapter 5
Factor the trinomials.
ODD RULE
2
Factor. 10 x + 7 x − 12
EVEN RULE ( even + even )
2
Factor. 8 x − 14 x − 15
EVEN RULE ( odd + odd )
2
Factor. 3 x − 34 x + 63
MAT 090/92
Chapter 5
MAT 090/92
Factor completely.
3x 2 − 10 x − 8
5x 2 + 7 x − 6
8 x 3 + 22 x 2 − 6 x
4 x 2 + 12 x + 5
6 x 2 + 4 y 2 − 19 xy
10 x 2 − 63x + 18
Chapter 5
MAT 090/92
Factor completely.
6 x 2 + 23x + 20
8x 2 + 6 x − 9
30 x 4 − 28 x 3 − 30 x 2
− 18 x 2 + 39 x − 18
6 x 2 + 17 xy − 28 y 2
12 x 2 − 28 x + 15
Chapter 5
MAT 090/92
Section 5.4
2
Factoring Perfect Square Trinomials in the form a x
2 2
2
Difference of Perfect Squares in the form a x − c .
2
+ 2acx + c 2 .
In both of these forms the first and last terms are perfect square numbers.
Perfect sq. numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 …
Factoring perfect square trinomials are created by squaring a binomial.
(ax + c )2 or (ax − c )2
(ax + c )2 = (ax + c )(ax + c ) = a 2 x 2 + acx + acx + c 2 = a 2 x 2 + 2acx + c 2
(ax − c )2 = (ax − c )(ax − c ) = a 2 x 2 − acx − acx + c 2 = a 2 x 2 − 2acx + c 2
Examples.
Factor completely.
x 2 + 10 x + 25
x 2 − 8 x + 16
x 2 + 10 x + 9
4 x 2 − 24 x + 36
16 x 2 + 24 x + 9
9 x 2 − 42 x + 49
Chapter 5
MAT 090/92
Factoring difference of perfect square binomials are created by FOILing two
binomials together, where the only difference are the signs.
(ax + c )(ax − c ) = a 2 x 2 − acx + acx − c 2 = a 2 x 2 − c 2
Be careful! Diff. of Per. Sq. can _______________________________________.
Examples.
Factor completely.
x 2 − 25
x 2 − 64
9 x 2 − 36
16 x 4 − 81
x2 + 4
12 x 2 − 75
− 36 x 2 + 25 y 2
100a 4 c 6 − 121x 8 y 2
Chapter 5
MAT 090/92
Section 5.5
3
3
Factoring Binomials that are Perfect Cubes in the form a ± b .
a 3 + b 3 ≠ (a + b )(a + b )(a + b )or (a + b )
3
(a + b )3 = a 3 + 3a 2b + 3ab 2 + b3 when FOILed!
Step 1. Perfect Cubes factor to a _____________ and ___________________.
Step 2. S.O.A.P the signs.
Step 3. Cube root the terms to build the binomial.
Step 4. SMILE your almost done!
Examples: Factor Completely.
27 y 3 + 8 x 3
64 x 3 − 125
x 6 y 3 − 216
1000 x 3 + 343
16ax 3 − 2at 3
64 x 6 − 1
Chapter 5
Section 5.6
Rules for factoring: Always looking to rewrite the term as multiplication.
Step 1. ALWAYS CHECK FOR A GCF!
Step 2. TWO terms are leftover.
Difference of Perfect SQUARES
PERFECT CUBES.
Step 3. THREE terms are leftover. ax 2 + bx + c
Step 4. FOUR terms are leftover.
MAT 090/92
Chapter 5
FACTOR COMPLETELY
5 x 4 − 80
3x 5 − 3x 3 − 24 x 2 + 24
7 x 2 + 35 x + 42
x2 + 6x + 9 − y2
MAT 090/92
4 x 4 + 10 x 3 − 36 x 2 − 90 x
x 6 − 64
3x 4 − 30 x 3 − 72 x 2
y2 − x2 − 6x − 9
Chapter 5
MAT 090/92
Section 5.7. Zero Product Rule for solving quadratic equations by factoring.
2
Quadratic equations are the trinomials set equal to zero. ax + bx + c = 0
The Zero Product Rule states that if (A)(B) = 0, then A = 0 or B = 0.
Examples.
5 x( x − 2 ) = 0
7( x + 8)(2 x − 3) = 0
Solve the equations by factoring.
x 2 − 10 x + 24 = 0
x 2 − 8 x + 16 = 0
2 x( x − 1)( x + 1) = 0
x 2 − 10 x − 24 = 0
x2 = 6x
4 x 2 = 25
Chapter 5
Solve the equations by factoring.
x 2 + 16 x + 63 = 0
MAT 090/92
x 2 + 12 x = 28
2 x 3 + 8 x 2 − 24 x = 0
6 x 2 − x − 35 = 0
8 x 2 + 30 x − 27 = 0
2 x 3 + 3 x 2 − 8 x − 12 = 0
(x + 3)(2 x − 1) = 9
(x − 3)(x + 4) = 5 x
Chapter 5
Section 5.8. Solving Word Problems.
MAT 090/92
Chapter 5
MAT 090/92
Chapter 5
A number is 6 less than its square. Find all such numbers.
MAT 090/92