Questions and Answers
What is the total number of roots of the polynomial function f(x) = 3x^6 + 2x^5 + x^4 - 2x^3?
6
What is the total number of roots of the polynomial function g(x) = 5x - 12x^2 + 3?
2
What is the total number of roots of the polynomial function f(x) = (3x^4 + 1)^2?
8
What is the total number of roots of the polynomial function g(x) = (x - 5)^2 + 2x^3?
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What is the total number of roots of the polynomial function f(x) = (x + 1)(x - 3)(x - 4)?
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What is the total number of roots of the polynomial function f(x) = (x - 6)^2(x + 2)^2?
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What is the total number of roots of the polynomial function f(x) = (x + 5)^3(x - 9)(x + 1)?
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What is the total number of roots of the polynomial function f(x) = (x + 2)(x - 1)[x - (4 + 3i)][x - (4 - 3i)]?
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What are the root(s) of f(x) = (x - 6)^2(x + 2)^2?
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What are the root(s) of f(x) = (x + 5)^3(x - 9)^2(x + 1)?
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What is 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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What is 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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What is 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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What is 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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At x = 2, the graph of f(x) = (x − 2)^3(x + 6)^2(x + 12) _______ the x-axis. At x = −6, the graph _______ the x-axis. At x = −12, the graph _______ the x-axis.
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At x = −4, the graph of g(x) = (x + 4)^4(x − 9) ______ the x-axis. At x = 9, the graph ______ the x-axis.
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If you know a root of a function is -2 + √3i, then _____. (Choose one)
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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? (Choose one)
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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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Using the fundamental theorem of algebra, what is the number and nature of the remaining root(s) for the polynomial function f(x) = x^3 - 7x - 6 if two roots are -2 and 3?
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Study Notes
Fundamental Theorem of Algebra
- The theorem states that a polynomial of degree ( n ) has exactly ( n ) roots, counting multiplicities and including complex roots.
Total Number of Roots
- For ( f(x) = 3x^6 + 2x^5 + x^4 - 2x^3 ): Total roots = 6
- For ( g(x) = 5x - 12x^2 + 3 ): Total roots = 2
- For ( f(x) = (3x^4 + 1)^2 ): Total roots = 8
- For ( g(x) = (x - 5)^2 + 2x^3 ): Total roots = 3
- For polynomial ( f(x) = (x + 1)(x - 3)(x - 4) ): Total roots = 3
- For ( f(x) = (x - 6)^2(x + 2)^2 ): Total roots = 4
- For ( f(x) = (x + 5)^3(x - 9)(x + 1) ): Total roots = 5
- For ( f(x) = (x + 2)(x - 1)[x - (4 + 3i)][x - (4 - 3i)] ): Total roots = 4
Roots with Multiplicities
- For ( f(x) = (x - 6)^2(x + 2)^2 ):
- Roots are ( 6 ) (multiplicity 2) and ( -2 ) (multiplicity 2).
- For ( f(x) = (x + 5)^3(x - 9)^2(x + 1) ):
- Roots are ( -5 ) (multiplicity 3), ( 9 ) (multiplicity 2), and ( -1 ) (multiplicity 1).
Number of x-Intercepts
- For ( f(x) = (x + 1)(x - 3)(x - 4) ): x-intercepts = 3
- For ( f(x) = (x - 6)^2(x + 2)^2 ): x-intercepts = 2
- For ( f(x) = (x + 5)^3(x - 9)(x + 1) ): x-intercepts = 3
- For ( f(x) = (x + 2)(x - 1)[x - (4 + 3i)][x - (4 - 3i)] ): x-intercepts = 2
Graph Behavior at Roots
- For ( f(x) = (x - 2)^3(x + 6)^2(x + 12) ):
- At ( x = 2 ), graph crosses x-axis.
- At ( x = -6 ), graph touches x-axis.
- At ( x = -12 ), graph crosses x-axis.
- For ( g(x) = (x + 4)^4(x - 9) ):
- At ( x = -4 ), graph touches x-axis.
- At ( x = 9 ), graph crosses x-axis.
Complex Roots
- If a root is ( -2 + \sqrt{3}i ), then ( -2 - \sqrt{3}i ) is a known root due to the complex conjugate theorem.
Roots of Specific Polynomials
- For ( f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 ):
- Known roots are ( -1, 1, 3 + i ).
- Nature of roots: 3 real roots and 2 imaginary roots.
- For ( g(x) = (x^2 + 3x - 4)(x^2 - 4x + 29) ):
- Roots include ( 1, -4, 2 + 5i, 2 - 5i ).
Remaining Roots Using Theorems
- For ( f(x) = x^3 - 7x - 6 ) with known roots ( -2 ) and ( 3 ):
- Degree of polynomial = 3, implying 3 total roots.
- With two known roots, one root remains and must be real due to complex conjugate theorem for imaginary pairs.
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Test your understanding of the Fundamental Theorem of Algebra with these flashcards. Each card challenges you to determine the total number of roots of various polynomial functions. Perfect for reviewing key concepts in algebra!
