9-5 Notes
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Transcript of 9-5 Notes
![Page 1: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/1.jpg)
Section 9-5The Factor Theorem
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Warm-up Let f(x) = 3x2 − 40x + 48.
1. Which of the following polynomials is a factor of f(x)?
e. x −12 a. x − 2 b. x − 3 c. x − 4 d. x − 6 f. x − 24
2. Which of the given values equals 0?
e. f(12) a. f(2) b. f(3) c. f(4) d. f(6) f. f(24)
![Page 3: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/3.jpg)
Warm-up Let f(x) = 3x2 − 40x + 48.
1. Which of the following polynomials is a factor of f(x)?
e. x −12
2. Which of the given values equals 0?
e. f(12) a. f(2) b. f(3) c. f(4) d. f(6) f. f(24)
![Page 4: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/4.jpg)
Warm-up Let f(x) = 3x2 − 40x + 48.
1. Which of the following polynomials is a factor of f(x)?
e. x −12
2. Which of the given values equals 0?
e. f(12)
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Solver
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Solver
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Solver
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Solver
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Solver
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Factor Theorem
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Factor Theorem
For a polynomial f(x), a number c is a solution to f(x) = 0 IFF (x - c) is a factor of f.
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Factor-Solution-Intercept Equivalence Theorem
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Factor-Solution-Intercept Equivalence Theorem
For any polynomial f, the following are all equivalent:
![Page 14: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/14.jpg)
Factor-Solution-Intercept Equivalence Theorem
(x - c) is a factor of f
For any polynomial f, the following are all equivalent:
![Page 15: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/15.jpg)
Factor-Solution-Intercept Equivalence Theorem
(x - c) is a factor of f
f(c) = 0
For any polynomial f, the following are all equivalent:
![Page 16: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/16.jpg)
Factor-Solution-Intercept Equivalence Theorem
(x - c) is a factor of f
f(c) = 0
c is an x-intercept of the graph y = f(x)
For any polynomial f, the following are all equivalent:
![Page 17: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/17.jpg)
Factor-Solution-Intercept Equivalence Theorem
(x - c) is a factor of f
f(c) = 0
c is an x-intercept of the graph y = f(x)
c is a zero of f
For any polynomial f, the following are all equivalent:
![Page 18: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/18.jpg)
Factor-Solution-Intercept Equivalence Theorem
(x - c) is a factor of f
f(c) = 0
c is an x-intercept of the graph y = f(x)
c is a zero of f
The remainder when f(x) is divided by (x - c) is 0
For any polynomial f, the following are all equivalent:
![Page 19: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/19.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others. 12x3 − 41x2 +13x + 6
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Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others. 12x3 − 41x2 +13x + 6
![Page 21: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/21.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
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Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3x2
4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
![Page 23: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/23.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3x2
−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
![Page 24: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/24.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3x2
−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
−44x2 +13x
![Page 25: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/25.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3x2
−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
−44x2 +13x
−11x
![Page 26: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/26.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3x2
−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
−44x2 +13x
−11x
−(−44x2 −11x)
![Page 27: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/27.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3x2
−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
−44x2 +13x
−11x
−(−44x2 −11x) 24x + 6
![Page 28: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/28.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3x2
−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
−44x2 +13x
−11x
−(−44x2 −11x) 24x + 6
+6
![Page 29: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/29.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3x2
−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
−44x2 +13x
−11x
−(−44x2 −11x) 24x + 6
+6
−(24x + 6)
![Page 30: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/30.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3x2
−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
−44x2 +13x
−11x
−(−44x2 −11x) 24x + 6
+6
−(24x + 6)
3x2 −11x + 6
![Page 31: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/31.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3x2
−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
−44x2 +13x
−11x
−(−44x2 −11x) 24x + 6
+6
−(24x + 6)
3x2 −11x + 6
(3x − 2)(x − 3)
![Page 32: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/32.jpg)
Example 1Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3x2
−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6
12x3 − 41x2 +13x + 6
−44x2 +13x
−11x
−(−44x2 −11x) 24x + 6
+6
−(24x + 6)
3x2 −11x + 6
(3x − 2)(x − 3)
(4x +1)(3x − 2)(x − 3)Answer:
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Example 2Find an equation for a polynomial function p with the zeros 2, -4,
and 4/7.
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Example 2Find an equation for a polynomial function p with the zeros 2, -4,
and 4/7.
p(x) = (x − 2)(x + 4)(7x − 4)
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Example 2Find an equation for a polynomial function p with the zeros 2, -4,
and 4/7.
p(x) = (x − 2)(x + 4)(7x − 4)
= (x2 + 2x − 8)(7x − 4)
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Example 2Find an equation for a polynomial function p with the zeros 2, -4,
and 4/7.
p(x) = (x − 2)(x + 4)(7x − 4)
= (x2 + 2x − 8)(7x − 4)
= 7x3 +10x2 − 64x + 32
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Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,
t(6) = 0, and t(-4/3) = 0.
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Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,
t(6) = 0, and t(-4/3) = 0.
t + 2
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Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,
t(6) = 0, and t(-4/3) = 0.
t + 2 t − 4
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Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,
t(6) = 0, and t(-4/3) = 0.
t + 2 t − 4 t − 6
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Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,
t(6) = 0, and t(-4/3) = 0.
t + 2 t − 4 t − 6
t +43
![Page 42: 9-5 Notes](https://reader033.fdocuments.us/reader033/viewer/2022042700/5559c72bd8b42a236c8b564e/html5/thumbnails/42.jpg)
Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,
t(6) = 0, and t(-4/3) = 0.
t + 2 t − 4 t − 6
t +43
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Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,
t(6) = 0, and t(-4/3) = 0.
t + 2 t − 4 t − 6
t +43 3t + 4
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Homework
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Homework
p. 587 #2-18