Chapter 8 Polynomials and Factoring
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1 Chapter 8 Polynomials and Factoring
2 8.1 Add and Subtract Polynomials
3 Monomial A. EX: Degree of a monomial the of all of the of the EX: 4x 2 y
4 Polynomial A or EX: Degree of a polynomial the of its terms EX: 2x 2 + 5x + 7
5 Leading coefficient the coefficient of the when the polynomial is written in EX: 2x 2 + 5x + 7
6 EX: Write the polynomial so that the exponents decrease from left to right. Identify the degree and the leading coefficient. 7 5y x 2 + x 3 7x
7 Binomial a polynomial with EX: Trinomial a polynomial with EX:
8 To add polynomials - Add REMEMBER: You can only add if the AND the are the same. EX: Find the sum. (6a 2 4) + (2a 2 9)
9 EX: Find the sum. (5x 3 + 4x 2x) + (4x 2 + 3x 3 6)
10 To subtract polynomials - Distribute the then. Make sure to when distributing the negative. EX: Find the difference. (4n 2 + 5) (-2n 2 + 2n 4)
11 EX: Find the difference. (4x 2 7x) (5x 2 + 4x 9)
12 EX: Write a polynomial that represents the perimeter of the figure. All sides added up.
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14 EX: Major League Baseball teams are divided into two leagues. During the period , the attendance (in thousands) at National (N) and American (A) games can be modeled by: N = -488t t + 24,700 A = -318t t + 25,600 Where t is the number of years since About how many total people attended games in 2001?
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17 8.2 Multiplying Polynomials
18 To multiply polynomials: everything in the to everything in the. REMEMBER: When you like bases, the exponents.
19 EX: Find the product. x(2x 3-7x 2 + 4) (x 2)(x 2 +2x + 1)
20 (3y 2 y + 5)(2y 3)
21 FOIL Method When multiplying, you can use the FOIL Method. The FOIL Method is the but in a. Multiply: Firsts Outers Inners Lasts
22 FOIL Method
23 EX: Find the product. (4b 5)(b 2) (6n 1)(n + 5)
24 (2x + 3y)2
25 EX: Simplify the expression. -3x 2 (x+ 11) (4x 5)(3x 2)
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27 EX: Write a polynomial that represents the area of the shaded region.
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30 EX: You are planning to build a walkway that surrounds a rectangular garden, as shown. The width of the walkway is the same on every side. Write a polynomial that represents the combined area of the garden and the walkway. Find the combined area when the width of the walkway is 4 feet. 10 ft x 9 ft
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33 8.4 Solve Polynomial Equations in Factored Form
34 Factoring To factor a polynomial To Factor a Polynomial: Step 1: Look for a A monomial that can be of in the polynomial Write the common monomial and by what is left over
35 EX: Factor out the greatest common monomial factor. 12x + 42y 4x x 3
36 15n 3 25n 8a 2 b 6ab 2 + 4ab
37 Zero-Product Property If ab = 0, then. This property is used to solve an equation when and the other side is a. EX: (x-3)(2x + 7) = 0
38 To solve an equation by factoring: 1) Put the equation in set equal to. EX: 2). 3) Set each factor and.
39 EX: Solve the equation. (2y + 5)(7y 5)= 0
40 a 2 + 5a = 0-28m 2 = 8m
41 Vertical Motion Model Models the of a An object thrown in the air with only the acting on it.
42 Vertical Motion Model h = -16t 2 + vt + s h = of the object (in feet) t = the object is in the air (in seconds) v = (in feet per second) s = of the object (in feet)
43 EX: A dolphin jumped out of the water with an initial vertical velocity of 32 feet per second. After how many seconds did the dolphin enter the water? After 1 second, how high was the dolphin above the water?
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45 8.5 Factor x 2 + bx + c
46 Review: Multiplying results in a. EX: (3x + 2)(x 4) Therefore, we will.
47 To factor a polynomial: Step 1: Look for a. Step 2: If you have a, factor it into. The 1 st terms in each binomial must to get the 1 st term in the trinomial. The 2 nd term in each binomial must to get the coefficient in the trinomial and must to get the in the trinomial. NOTE: Pay attention to.
48 NOTEBOOK EXAMPLE #1 EX: Factor the trinomial. x 2 + 3x + 2 t 2 + 9t + 14 x 2 4x + 3 t 2 8t + 12 m 2 + m 20 w 2 + 6w 16 x 2 4xy + 4y 2 m 2 mn 42n 2
49 NOTEBOOK EXAMPLE #2 EX: Solve the equation (by factoring). x 2 2x = 24 x 2 2x 8 = 7 s(s + 1) = 72
50 EX: Find the dimensions of the rectangle. 8.5 Ex #43
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53 EX: You are designing a flag for the SMCC football team with the dimensions shown. The flag requires 80 square feet of fabric. Find the width w of the flag. SMCC
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56 8.6 Factor ax 2 + bx + c
57 Steps to Factoring: 1) Look for a. If the leading coefficient is, factor out a. 2) If you have a, factor it into.
58 Test to factor a trinomial when the leading coefficient is : In your two binomials: The 1 st terms must to get the in the trinomial. The 2 nd terms must to get the in the trinomial. plus must equal the in the trinomial.
59 NOTEBOOK EXAMPLE #3 EX: Factor the trinomial. 3x 2 + 8x + 4 4x 2 9x + 5 2x x 7-2x 2 5x 3-5m 2 + 6m 1-4n 2 16n 15
60 NOTEBOOK EXAMPLE #4 EX: Solve the equation (by factoring). 8x 2 2x = 3 b(20b 3) = 2 6x 2 15x = 99
61 EX: In a shot put event, an athlete throws the shot put from an initial height of 6 feet and with an initial vertical velocity of 29 feet per second. After how many seconds does the shot put hit the ground?
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64 EX: The length of a rectangle is 7 inches more than 5 times its width. The area of the rectangle is 6 square inches. What is the width?
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67 8.7 Factoring Special Products
68 Steps to Factoring: 1). 2) Trinomial into. If the two binomials are the, its called a. 3) If you have a : Factor it into with the but. EX: x 2 9
69 NOTEBOOK EXAMPLE #5 EX: Factor the polynomial. y x x 2 81y x 2-4x x 2 12x x 2 + 4xy + y 2-3y y 108
70 NOTEBOOK EXAMPLE #6 EX: Solve the equation. 4x 2 36 = 0-8k 2 = -98 y y =
71 EX: A window washer drops a wet sponge from a height of 64 feet. After how many seconds does the sponge land on the ground?
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74 8.8 Factor Polynomials Completely
75 Steps to Factoring: 1). 2) Trinomial into. 3) Difference of square. Same, different. 4) If you have a, factor by : Group the polynomial in. Factor out a from each. Factor out a from each pair.
76 NOTEBOOK EXAMPLE #7 EX: Factor. x(x- 2) + (x 2) 6z(z 4) 7(z 4) 12(z 1) 5z 2 (1 z) a 3 + 3a 2 + a + 3 a a 2 5a 65
77 NOTEBOOK EXAMPLE #8 EX: Factor completely. 3x 3 12x 2y 3 12y y 7a 3 b 3 63ab 3
78 NOTEBOOK EXAMPLE #9 EX: Solve the equation. w 3 8w w = 0 x 4 25x 2 = 0 4y 3 7y 2 = 16y - 28
79 EX: A terrarium has a volume of 4608 cubic inches. Its length is more than 10 inches. Its dimensions are shown. Find the length, width, and height of the terrarium.
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Section 9.1: Add and Subtract Polynomials. The number part of a term with a variable part.
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