Q. 5.90 - Intermediate Algebra

Question

Use the Remainder Theorem to find the remainder when $f \left(x\right) = x^3 - 7x + 12$ is divided by $x + 3$.

Step-by-Step Solution

Answer

The remainder is 6 when $f \left(x\right) = x^3 - 7x + 12$ is divided by $x+3$.

Step 1. Given Information

We want to find the remainder when $x^3-7x+12$ is divided by $x+3$

We know that To use the Remainder Theorem, we must use the divisor in the x − c form.

We can write the divisor $x + 3$ as $x - (-3).$

So, our c is −3.

To find the remainder, we evaluate f (c) which is f (−3).

Step 2. Evaluate f (−3)

To evaluate f (−3), substitute x = −3 and simplify. 

$$\begin{gathered} f \left(x\right) = x^3 - 7x + 12 \\ f (-3) = {(-3)}^3 - 7(-3) + 12 \\ f(-3) =-27+21+12 \\ f(-3) = 6 \end{gathered}$$

The remainder is 6 when $f \left(x\right) = x^3 - 7x + 12$ is divided by $x + 3$.

Step 3. Check:

Use synthetic division to check.

The remainder is 6.