The Fundamental Theorem of Algebra

g(x) = 5x - 12x2 + 3

2

g(x) = (x - 5)2 + 2x3

3

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

D,F

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

6

f (x) = (3x4 + 1)2

8

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

A,D,F

Identify all of the root(s) of g(x) = (x2 + 3x - 4)(x2 - 4x + 29).

B,C,E,F

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?

D

Describe the graph of the function at its roots. f(x) = (x − 2)3(x + 6)2(x + 12)

crosses touches crosses

g(x) = (x + 4)4(x − 9)

touches crosses

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.

Answer: The degree of the polynomial is 3. By the fundamental theorem of algebra, the function has three 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. x=1