5. Consider the function $$ f(x)=x^{6}-2 x^{3}+5 x+1 $$ and answer the following questions. a. (2 pts) State the Remainder Theorem. b. (6 pts) Use the Remainder Theorem to determine the remainder when $$ f(x) $$ is divided by $$ x+2 $$. Don't forget to show your work! No credit will be given if you don't use the Remainder Theorem.

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UpStudy AI · Accepted Answer

**a.** According to the Remainder Theorem, if a polynomial $$ f(x) $$ is divided by $$(x - a)$$, then the remainder is $$ f(a) $$.

**b.** Here, the divisor is $$ x+2 $$, which can be written as $$ x - (-2) $$. By the Remainder Theorem, the remainder is $$ f(-2) $$.

We calculate:
$$
f(-2)=(-2)^6-2(-2)^3+5(-2)+1
$$
Step by step:

1. Compute $$(-2)^6$$:
$$
(-2)^6=64
$$

2. Compute $$(-2)^3$$:
$$
(-2)^3=-8
$$

3. Multiply by $$-2$$:
$$
-2(-8)=16
$$

4. Compute $$5(-2)$$:
$$
5(-2)=-10
$$

Now add all the terms:
$$
f(-2)=64+16-10+1
$$
$$
f(-2)=71
$$

Thus, the remainder when $$ f(x) $$ is divided by $$ x+2 $$ is $$ 71 $$.
**a.** The Remainder Theorem states that when a polynomial $$ f(x) $$ is divided by $$ (x - a) $$, the remainder is $$ f(a) $$. **b.** Using the Remainder Theorem, the remainder when $$ f(x) = x^6 - 2x^3 + 5x + 1 $$ is divided by $$ x + 2 $$ is $$ f(-2) = 71 $$.

Consider the function and answer the
following questions.
a. (2 pts) State the Remainder Theorem.
b. (6 pts) Use the Remainder Theorem to determine the remainder
when is divided by . Don’t forget to show your work!
No credit will be given if you don’t use the Remainder
Theorem.

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Consider the function and answer the following questions. a. (2 pts) State the Remainder Theorem. b. (6 pts) Use the Remainder Theorem to determine the remainder when is divided by . Don’t forget to show your work! No credit will be given if you don’t use the Remainder Theorem.