Questions and Answers
Determine the total number of roots of the polynomial function f(x) = 3x^6 + 2x^5 + x^4 - 2x^3.
6
Determine the total number of roots of the polynomial function g(x) = 5x - 12x^2 + 3.
2
Determine the total number of roots of the polynomial function f(x) = (3x^4 + 1)^2.
8
Determine the total number of roots of the polynomial function g(x) = (x - 5)^2 + 2x^3.
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Determine the total number of roots of the polynomial function using the factored form f(x) = (x + 1)(x - 3)(x - 4).
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Determine the total number of roots of the polynomial function using the factored form f(x) = (x - 6)^2(x + 2)^2.
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Determine the total number of roots of the polynomial function using the factored form f(x) = (x + 5)^3(x - 9)(x + 1).
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Determine the total number of roots of the polynomial function using the factored form f(x) = (x + 2)(x - 1)[x - (4 + 3i)][x - (4 - 3i)].
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Find the root(s) of f(x) = (x - 6)^2(x + 2)^2.
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Find the root(s) of f(x) = (x + 5)^3(x - 9)^2(x + 1).
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Determine the number of x-intercepts that appear on the graph of the function f(x) = (x + 1)(x - 3)(x - 4).
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Determine the number of x-intercepts that appear on the graph of the function f(x) = (x - 6)^2(x + 2)^2.
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Determine the number of x-intercepts that appear on the graph of the function f(x) = (x + 5)^3(x - 9)(x + 1).
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Determine the number of x-intercepts that appear on the graph of the function f(x) = (x + 2)(x - 1)[x - (4 + 3i)][x - (4 - 3i)].
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Describe the graph of the function at its roots f(x) = (x - 2)^3(x + 6)^2(x + 12): At x = 2, the graph _______ the x-axis.
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Describe the graph of the function at its roots f(x) = (x - 2)^3(x + 6)^2(x + 12): At x = -6, the graph ______ the x-axis.
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Describe the graph of the function at its roots f(x) = (x - 2)^3(x + 6)^2(x + 12): At x = -12, the graph ______ the x-axis.
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G(x) = (x + 4)^4(x - 9): At x = -4, the graph ______ the x-axis.
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G(x) = (x + 4)^4(x - 9): At x = 9, the graph ______ the x-axis.
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If you know a root of a function is -2 + \sqrt{3}i, then _____.
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Three roots of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 are -1, 1, and 3 + i. Which of the following describes the number and nature of all the roots of this function?
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Identify all of the root(s) of g(x) = (x^2 + 3x - 4)(x^2 - 4x + 29).
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Use the fundamental theorem of algebra and the complex conjugate theorem to determine the number and nature of the remaining root(s) for f(x) = x^3 - 7x - 6, given that two roots are -2 and 3.
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Study Notes
Fundamental Theorem of Algebra
- The total number of roots of a polynomial function equals its degree.
Polynomial Functions and Their Roots
- For the polynomial function f(x) = 3x^6 + 2x^5 + x^4 - 2x^3, there are 6 roots.
- For g(x) = 5x - 12x^2 + 3, the total number of roots is 2.
- The function f(x) = (3x^4 + 1)^2 has 8 roots.
- For g(x) = (x - 5)^2 + 2x^3, there are a total of 3 roots.
Factored Form and Roots
- The polynomial f(x) = (x + 1)(x - 3)(x - 4) has 3 roots.
- For f(x) = (x - 6)^2(x + 2)^2, the total number of roots is 4.
- The function f(x) = (x + 5)^3(x - 9)(x + 1) has 5 roots.
- For f(x) = (x + 2)(x - 1)[x - (4 + 3i)][x - (4 - 3i)], it contains 4 roots.
Identifying Specific Roots
- The roots of f(x) = (x - 6)^2(x + 2)^2 include 6 (multiplicity 2) and -2 (multiplicity 2).
- For f(x) = (x + 5)^3(x - 9)^2(x + 1), the roots are -5 (multiplicity 3), 9 (multiplicity 2), and -1 (multiplicity 1).
X-intercepts on Graphs
- The function f(x) = (x + 1)(x - 3)(x - 4) has 3 x-intercepts.
- For f(x) = (x - 6)^2(x + 2)^2, there are 2 x-intercepts.
- The function f(x) = (x + 5)^3(x - 9)(x + 1) also has 3 x-intercepts.
- For f(x) = (x + 2)(x - 1)[x - (4 + 3i)][x - (4 - 3i)], there are 2 x-intercepts.
Graph Behavior at Roots
- In f(x) = (x − 2)^3(x + 6)^2(x + 12):
- At x = 2, the graph crosses the x-axis.
- At x = -6, the graph touches the x-axis.
- At x = -12, the graph crosses the x-axis.
- For g(x) = (x + 4)^4(x − 9):
- At x = -4, the graph touches the x-axis.
- At x = 9, the graph crosses the x-axis.
Complex Conjugate Theorem
- If a root of the function is -2 + sqrt(3)i, the corresponding known root is -2 - sqrt(3)i.
Analysis of Roots in a Polynomial
- For f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20, having roots -1, 1, and 3 + i:
- The function has three real roots and two imaginary roots.
Identifying Roots from a Polynomial
- For g(x) = (x^2 + 3x - 4)(x^2 - 4x + 29), the roots include 1, -4, and two non-real roots: 2 + 5i and 2 - 5i.
Remaining Roots Calculation
- For the polynomial function f(x) = x^3 − 7x − 6, with known roots -2 and 3:
- The degree of the polynomial is 3, indicating 3 roots total.
- Since two roots are given, one additional real root remains, inferred from the complex conjugate theorem which requires any imaginary roots to appear in pairs.
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Description
This quiz focuses on the Fundamental Theorem of Algebra, testing your knowledge on determining the total number of roots for various polynomial functions. Each flashcard presents a polynomial function, and you need to identify the correct number of roots. Sharpen your algebra skills and deepen your understanding of polynomial equations!