Transcript Warm Up
Chapter 10
Adding and Subtracting Polynomials
CHAPTER 10.1
Vocabulary
Polynomial
◦ Expression whose terms are of the form
where k is a nonnegative integer.
Standard form
2 x 5x 4 x 7
3
Degree
2
◦ Exponent of the variable for each term
Degree of a polynomial
◦ The largest degree of its terms
Leading coefficient
◦ The coefficient of the first term
ax
k
Classifying Polynomials
Polynomial
Degree Classified
by degree
6
0
Constant
Classified by
number of
terms
Monomial
-2x
1
Linear
Monomial
3x + 1
1
Linear
Binomial
-x² + 2x – 5
2
Quadratic
Trinomial
4x³ - 8x
3
Cubic
Binomial
2x 4 - 7x³ - 5x + 1
4
Quartic
Polynomial
Adding Polynomials
1)
2x
2
x 5 x x 6
2
2x x 5 x x 6
2
2
3x 2 x 1
2
x 2x 7) (3x 7 4x) (4x 8 x )
3
2
2
2
3
5 x x 2 x 7 3x 7 4 x 4 x 8 x
3
2
4 x 9 x 5x 6
2) (5x
3
2
2
2
3
Subtracting Polynomials
1)
2x
3
5x x 8 2x 3x 4
2
3
2 x 5 x x 8 2 x 3x 4
3
2
3
5x 4 x 12
2
2
2) ( x 8) (7 x 4x )
2
2
x 8 7 x 4x
2
3x 7 x 8
2
2
3) (3x 5x 3) (2 x x 4)
2
2
3x 5 x 3 2 x x 4
x2 4x 7
2
Adding and Subtracting Polynomials
(9x x 7 x) ( x 6x 2x 9) (4x 3x 8)
4
2
3
2
3
9 x x 7 x x 6 x 2 x 9 4 x 3x 8
4
2
3
2
9 x 3x 7 x 6 x 17
4
3
2
3
Chapter 10.2
Multiplying Polynomials
Multiply the Polynomials
Use the distributive property
2) x(9 x 4 x 3)
2
1) (12x)(12x 11)
(12x)(12x) (12x)(11)
144x 132x
2
3) (2 x 7 y)(8xy)
(2 x)(8xy) (7 y)(8xy)
16x y 56xy
2
2
x(9 x ) x(4 x) x(3)
9 x 3 4 x 2 3x
2
4) 11xy(2 x 3 y 2 )
11xy(2 x) (11xy)(3 y )
2
22x y 33xy
2
3
Multiply the Polynomials
Use the distributive property
9x(3x 9x 11)
2
(9x)(3x ) (9x)(9x) (9x)(11)
27x 3 81x 2 99x
5)
2
6) (11x)(5x 8x 9 x 8)
3
2
(11x)(5x ) (11x)(8x ) (11x)(9x) (11x)(8)
55x 4 88x 3 99x 2 88x
3
2
(x + 2)(x – 3)
x
x
2
x²
2x
-3x
-3
-6
x² - x – 6
(3x + 4)(x + 5)
3x
4
x 3x² 4x
5 15x 20
3x² + 19x +20
(3x + 4)(2x + 1)
3x
4
2x 6x² 8x
1
3x 4
6x² + 11x +4
(3x + 10)(2x + 6)
3x
10
2x 6x² 20x
18x
60
6
6x² + 38x +60
(4x² - 3x – 1)(2x – 5)
2x -5
4x² 8x³ -20x²
-3x -6x² 15x
-1 -2x 5
8x³ - 26x² + 13x + 5
(x – 2)(5 + 3x - x²)
x -2
5 5x -10
3x 3x²
-6x
-x² -x³ 2x²
-x³ + 5x² - x – 10
Special Products of Polynomials
CHAPTER 10.3
(x + 3)² = (x + 3)(x + 3)
x
x
3
x²
3x
3x
3
9
x² + 6x + 9
(3x + 4)² = (3x + 4)(3x + 4)
3x
4
3x 9x² 12x
12x
16
4
9x² + 24x + 16
(x – 2)² = (x – 2)(x – 2)
x
x
-2
x²
-2x
-2x
-2
4
x² - 4x + 4
(2x – 7y)² = (2x – 7y)(2x – 7y)
2x
-7y
2x 4x² -14xy
-14xy
49y²
-7y
4x² - 28x + 49y²
Dividing Polynomials
CHAPTER 11.7
1) Divide 12x² - 20x + 8 by 4x.
12x 20x 8 12x
20x 8
4x
4x
4x 4x
2
2
2) Divide 8x + 14 by 2.
8 x 14
2
8 x 14 4 x 7
2
2
3) Divide 9c² + 3c by c.
9c 3c 9c
3c
9c 3
c
c
c
2
2
2
3x 5
x
4) Divide -2x² - 12x by -2x.
2 x 12x 2 x
12x
x6
2x
2x 2x
2
2
5) Divide 9a² -54a – 36 by 3a.
9a 54a 36 9a 54a 36
3a
3a
3a 3a
12
3a 18
a
2
2
Factoring x² + bx + c
CHAPTER 10.5
Factoring
To factor x² + bx + c you need to find
numbers p and q such that
p+q=b
and
pq = c
x² + bx + c = (x + p)(x + q) when p + q = b and pq = c
Example:
x² + 6x + 8 = (x + 4)(x + 2)
4 + 2 = 6 and 4(2) = 8
Factor x² + 3x + 2
Find the factors of 2
1
2
-1
-2
(x + 1)(x + 2)
Factor x² - 5x + 6
Find the factors of 6
1
6
-1
-6
2
3
-2
-3
(x – 2)(x – 3)
Factor x² - 2x – 8
Find the factors of -8
1
-8
-1
8
2
-4
-2
4
(x + 2)(x – 4)
Factor x² + 7x – 18
Find the factors of -18
1
-18
-1
18
2
-9
-2
9
3
-6
-3
6
(x – 2)(x + 9)
Factoring ax² + bx + c
CHAPTER 10.6
Factor 2x²
2x² + 11x + 5
x
2(5) = 10
1
10
-1
2
-10
5
-2
-5
2x
1
5
10x
x
(x + 5)(2x + 1)
Factor 3x²
3x² - 4x –- 7
3x
3(-7) = -21
1
-21
-1
21
3
-7
-3
7
x
1
-7
-7x
3x
(3x – 7)(x + 1)
Factor 6x²
6x² - 19x + 15
15
6(15) = 90
1 90
-1 -90
2 45
-2 -45
30
3
6 15 -3 -30
-6 -15 5
18
9 10 -5 -18
-9 -10
3x
2x
-3
-5
-10x
-9x
(3x – 5)(2x – 3)
Factor 6x² - 2x – 8
2(3x²
3x² -x –- 44) 3x
3(-4) = -12
1 -12
-1 12
2
-6
-2
6
3 -4
-3 4
x
1
-4
-4x
3x
2(3x – 4)(x + 1)
Factoring Special Products
CHAPTER 10.7
Factoring Special Products
Difference of Two Squares
◦ a² - b² = (a + b)(a – b)
◦ 9x² - 16 = (3x + 4)(3x – 4)
Perfect Square Trinomial
◦
◦
◦
◦
a² + 2ab + b² = (a + b) ²
x² + 8x + 16 = (x + 4) ²
a² - 2ab + b² = (a – b) ²
x² - 12x + 36 = (x – 6) ²
Difference of Two Squares
1.
m² - 4
(m + 2)(m – 2)
2.
4p² - 25
(2p + 5)(2p – 5)
3.
50 – 98x²
2(25 – 49x²)
2(5 + 7x)(5 – 7x)
Perfect Square Trinomial
1.
x² - 4x + 4
(x – 2)²
2.
16y² + 24y + 9
(4y + 3)²
3.
3x² - 30x + 75
3(x² - 10x + 25)
3(x – 5)²
Factoring Using the Distributive Property
CHAPTER 10.8
Factoring Completely
1.
Find the GCF
2.
Factor out the GCF
3.
Factor the remaining terms
Practice
1) 14x 21x
4
2
7 x (2 x 3)
2
2
4) 4 x 3 20x 2 24x
4x( x 5x 6)
4 x( x 2)(x 3)
2
2
2
x
8
2)
2
2( x 4)
3) 2 x 2 8
2( x 4)
2( x 4)(x 4)
2
5) 45x 4 20x 2
5x (9 x 4)
2
5x (3x 2)(3x 2)
2
2
Factor by Grouping
1.
Group terms
2.
Factor each group
3.
Use distributive property
Practice
1) 6( x 1) 7( x 1)
2) 2 x( x 4) 7( x 4)
(2 x 7)(x 4)
(6 7)(x 1)
13( x 1)
3) x3 2 x 2 3x 6
( x 2x ) (3x 6)
2
x ( x 2) 3( x 2)
2
( x 3)(x 2)
3
2
4) x 3 2 x 2 9 x 18
( x 2x ) (9x 18)
2
x ( x 2) 9( x 2)
3
2
( x 9)(x 2)
2