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


 x6
 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