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Questions and Answers
Find the zeros of f(x) = 2x² - 11x + 5.
Find the zeros of f(x) = 2x² - 11x + 5.
5, 0.5
Find the zeros of f(x) = 2x³ - 9x² - 11x + 60.
Find the zeros of f(x) = 2x³ - 9x² - 11x + 60.
-2.5, 3, 4
Find the zeros of f(x) = 2x⁴ - x³ - 2x² + x.
Find the zeros of f(x) = 2x⁴ - x³ - 2x² + x.
0, 0.5, 1, -1
Find the zeros of f(x) = x³ - 2x² + x.
Find the zeros of f(x) = x³ - 2x² + x.
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Find the zeros of f(x) = x³ - 4x² - 5x.
Find the zeros of f(x) = x³ - 4x² - 5x.
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Find the zeros of f(x) = 3x³ - 5x² - 11x - 3.
Find the zeros of f(x) = 3x³ - 5x² - 11x - 3.
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Find the zeros of f(x) = 4x⁴ - 4x³ - 39x² + 20x + 100.
Find the zeros of f(x) = 4x⁴ - 4x³ - 39x² + 20x + 100.
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Find the zeros of f(x) = x² - 6x + 9.
Find the zeros of f(x) = x² - 6x + 9.
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Find the zeros of f(x) = x³ - 4x.
Find the zeros of f(x) = x³ - 4x.
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Find the zeros of f(x) = x³ + 4x² - 3x - 18.
Find the zeros of f(x) = x³ + 4x² - 3x - 18.
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Find the zeros of f(x) = 3x⁴ - 24x³ + 384x - 768.
Find the zeros of f(x) = 3x⁴ - 24x³ + 384x - 768.
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Find the zeros of f(x) = 9x² - 3x - 2.
Find the zeros of f(x) = 9x² - 3x - 2.
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Study Notes
Finding Zeros of Polynomial Functions
- Zeros of a polynomial function are the x-values where f(x) = 0. These points reflect the roots or intersections with the x-axis.
Example Polynomial Functions and Their Zeros
-
f(x) = 2x² - 11x + 5
- Zeros are approximately 5 and 0.5.
-
f(x) = 2x³ - 9x² - 11x + 60
- Zeros include -2.5, 3, and 4.
-
f(x) = 2x⁴ - x³ - 2x² + x
- Zeros are 0, 0.5, 1, and -1.
-
f(x) = x³ - 2x² + x
- Zeros are 0 and 1.
-
f(x) = x³ - 4x² - 5x
- Zeros found at -1, 0, and 5.
-
f(x) = 3x³ - 5x² - 11x - 3
- Zeros include 3, -0.333, and -1.
-
f(x) = 4x⁴ - 4x³ - 39x² + 20x + 100
- Zeros are approximately 2.5 and -2.
-
f(x) = x² - 6x + 9
- A double root at 3.
-
f(x) = x³ - 4x
- Zeros located at -2, 0, and 2.
-
f(x) = x³ + 4x² - 3x - 18
- Roots are 2 and -3.
-
f(x) = 3x⁴ - 24x³ + 384x - 768
- Zeros found at -4 and 4.
-
f(x) = 9x² - 3x - 2
- Zeros are approximately -0.333 and 0.667.
Summary of Key Concepts
- Finding zeros involves setting the polynomial equal to zero and solving for x.
- Zeros can be real or complex and may occur multiple times (multiplicity).
- The values of zeros can be found using algebraic methods such as factoring, synthetic division, or the quadratic formula, depending on the polynomial degree.
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