Question

Pls Help!!! worth 40 pts.

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 1

Your function is . The fundamental theorem of algebra says that there will be three roots, since the degree of the polynomial is 3. The problem provides two real roots, x = -2 and x = 3, so there must be one more.

The theorem also says that possible roots of the polynomial would be in this case, the factors of the constant (-6) over the factors of the coefficient of the term with the highest degree (1).

Factors of -6 are: 1, 2, 3, 6, -1, -2, -3, -6

Factors of 1 are: 1, -1

Possible rational roots are: 1, 2, 3, 6, -1, -2, -3, -6

I then use synthetic division to see which possible rational root is a real root by dividing by the possible rational roots, and I get a root when the remainder is 0. Remember to put the placeholder of 0 for x^2 when dividing:

-1} 1 0 -7 -6

-1 1 6

-----------------

1 -1 -6 0

When I divide by the possible rational root of -1, I get a remainder of 0, which means -1 is a root.

To check:

(x + 2)(x - 3)(x + 1)

= (x^2 - x - 6)(x + 1)

= x^3 - x^2 - 6x + x^2 - x - 6

= x^3 - 7x - 6

Explanation:

Answer 2

Your function is
`f(x) = x^3 - 7x - 6`. The fundamental theorem of algebra says that there will be three roots, since the degree of the polynomial is 3. The problem provides two real roots, x = -2 and x = 3, so there must be one more.

The theorem also says that possible roots of the polynomial would be in this case, the factors of the constant (-6) over the factors of the coefficient of the term with the highest degree (1).

Factors of -6 are: 1, 2, 3, 6, -1, -2, -3, -6
Factors of 1 are: 1, -1

Possible rational roots are: 1, 2, 3, 6, -1, -2, -3, -6

I then use synthetic division to see which possible rational root is a real root by dividing
`x^3 -7x -6` by the possible rational roots, and I get a root when the remainder is 0. Remember to put the placeholder of 0 for x^2 when dividing:
-1} 1 0 -7 -6
-1 1 6
-----------------
1 -1 -6 0

When I divide by the possible rational root of -1, I get a remainder of 0, which means -1 is a root.

To check:
(x + 2)(x - 3)(x + 1)
= (x^2 - x - 6)(x + 1)
= x^3 - x^2 - 6x + x^2 - x - 6
= x^3 - 7x - 6