The Fundamental Theorem of Algebra
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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.

<p>3</p> Signup and view all the answers

Determine the total number of roots of the polynomial function using the factored form f(x) = (x + 1)(x - 3)(x - 4).

<p>3</p> Signup and view all the answers

Determine the total number of roots of the polynomial function using the factored form f(x) = (x - 6)^2(x + 2)^2.

<p>4</p> Signup and view all the answers

Determine the total number of roots of the polynomial function using the factored form f(x) = (x + 5)^3(x - 9)(x + 1).

<p>5</p> Signup and view all the answers

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)].

<p>4</p> Signup and view all the answers

Find the root(s) of f(x) = (x - 6)^2(x + 2)^2.

<p>-6 with multiplicity 2</p> Signup and view all the answers

Find the root(s) of f(x) = (x + 5)^3(x - 9)^2(x + 1).

<p>-5 with multiplicity 3</p> Signup and view all the answers

Determine the number of x-intercepts that appear on the graph of the function f(x) = (x + 1)(x - 3)(x - 4).

<p>3</p> Signup and view all the answers

Determine the number of x-intercepts that appear on the graph of the function f(x) = (x - 6)^2(x + 2)^2.

<p>2</p> Signup and view all the answers

Determine the number of x-intercepts that appear on the graph of the function f(x) = (x + 5)^3(x - 9)(x + 1).

<p>3</p> Signup and view all the answers

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)].

<p>2</p> Signup and view all the answers

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.

<p>crosses</p> Signup and view all the answers

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.

<p>touches</p> Signup and view all the answers

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.

<p>crosses</p> Signup and view all the answers

G(x) = (x + 4)^4(x - 9): At x = -4, the graph ______ the x-axis.

<p>touches</p> Signup and view all the answers

G(x) = (x + 4)^4(x - 9): At x = 9, the graph ______ the x-axis.

<p>crosses</p> Signup and view all the answers

If you know a root of a function is -2 + \sqrt{3}i, then _____.

<p>-2 - \sqrt{3}i is a known root.</p> Signup and view all the answers

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?

<p>f(x) has three real roots and two imaginary roots.</p> Signup and view all the answers

Identify all of the root(s) of g(x) = (x^2 + 3x - 4)(x^2 - 4x + 29).

<p>-1</p> Signup and view all the answers

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.

<p>The degree of the polynomial is 3. By the Fundamental Theorem of Algebra, the function has 3 roots. Two roots are given, so there must be one root remaining. By the Complex Conjugate Theorem, imaginary roots come in pairs. The final root must be real.</p> Signup and view all the answers

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!

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