The Fundamental Theorem of Algebra-Assignment

Find the root(s) of f (x) = (x + 5)3(x - 9)2(x + 1). -5 with multiplicity 3 5 with multiplicity 3 -9 with multiplicity 2 9 with multiplicity 2 -1 with multiplicity 0 -1 with multiplicity 1 1 with multiplicity 0 1 with multiplicity 1

-5 with multiplicity 3 9 with multiplicity 2 -1 with multiplicity 1

Identify all of the root(s) of g(x) = (x2 + 3x - 4)(x2 - 4x + 29). -1 1 -4 4 2 + 5i 2 - 5i -2 + 10i -2 - 10i

1 -4 2+5i 2-5i

Determine the number of x-intercepts that appear on a graph of each function. f (x) = (x + 2)(x - 1)[x - (4 + 3i )][x - (4 - 3i)]

2

Determine the total number of roots of each polynomial function. f (x) = 3x6 + 2x5 + x4 - 2x3

6

g(x) = (x + 4)4(x − 9) At x = −4, the graph______ the x-axis. At x = 9, the graph ______ the x-axis.

touches crosses

Determine the total number of roots of each polynomial function. g(x) = 5x - 12x2 + 3

2

Two roots of the polynomial function f(x) = x3 − 7x − 6 are −2 and 3. Use the fundamental theorem of algebra and the complex conjugate theorem to determine the number and nature of the remaining root(s). Explain your thinking.

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.

f (x) = x5 − 8x4 + 21x3 − 12x2 − 22x + 20 Three roots of this polynomial function are −1, 1, and 3 + i. Which of the following describes the number and nature of all the roots of this function? A. f (x) has two real roots and one imaginary root. B. f (x) has three real roots. C. f (x) has five real roots. D. f (x) has three real roots and two imaginary roots.

D. f (x) has three real roots and two imaginary roots.

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. At x = −6, the graph ______ the x-axis. At x = −12, the graph ______ the x-axis.

crosses touches crosses

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

2

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

3

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

3

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

3

Determine the total number of roots of each polynomial function. g(x) = (x - 5)2 + 2x3

3

Determine the total number of roots of each polynomial function using the factored form. f (x) = (x + 2)(x - 1)[x - (4 + 3i)][x - (4 - 3i)]

4

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

4

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

5

Find the root(s) of f (x) = (x- 6)2(x + 2)2. -6 with multiplicity 1 -6 with multiplicity 2 6 with multiplicity 1 6 with multiplicity 2 -2 with multiplicity 1 -2 with multiplicity 2 2 with multiplicity 1 2 with multiplicity 2

6 with multiplicity 2 -2 with multiplicity 2

Determine the total number of roots of each polynomial function. f (x) = (3x4 + 1)2

8

Choose the answer that best completes the sentence. If you know a root of a function is -2+sqrt 3i, then _____. A.2+sqrt 3i is a possible root. B.2+sqrt 3i is a known root. C.-2-sqrt 3i is a possible root. D.-2-sqrt 3i is a known root.

D. -2-sqrt 3i is a known root