Chapter 4: Q. 20 (page 209)

Use the Remainder Theorem to find the remainder when $f (x)$ is divided by $x - c$ . Then use the Factor Theorem to determine whether $x - c$ is a factor of $f (x)$ .
$f (x) = 3 x^{4} + x^{3} - 3 x + 1; x + \frac{1}{3}$

Short Answer

Expert verified

The remainder is $2$ and $x + \frac{1}{3}$ is not a factor.

Step by step solution

Step 1. Given information.

The given function is,

$f (x) = 3 x^{4} + x^{3} - 3 x + 1; x + \frac{1}{3}$

Step 2. Using remainder theorem.

We know that if $f (x)$ is divided by $x - c$ then the remainder will be $f (c)$ .

Here $f (x)$ is divided by $x + \frac{1}{3}$ , then the remainder is $f (- \frac{1}{3})$ that is

$f (- \frac{1}{3}) = 3 {(- \frac{1}{3})}^{4} + {(- \frac{1}{3})}^{3} - 3 (- \frac{1}{3}) + 1 = \frac{1}{27} - \frac{1}{27} + 1 + 1 = 2$

Therefore, the remainder is $2$ .

Step 3. Using factor theorem.

We know that if $x - c$ is the factor of the polynomial $f (x)$ then

$f (c) = 0$ ,

Here $c = - \frac{1}{3}$ , then

$f (c) = f (- \frac{1}{3}) = 2$

We observe that $f (- \frac{1}{3}) \neq 0$ ,then $x + \frac{1}{3}$ is not a factor.

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Use the Remainder Theorem to find the remainder when $f (x)$ is divided by $x - c$ . Then use the Factor Theorem to determine whether $x - c$ is a factor of $f (x)$ .
$f (x) = 3 x^{4} + x^{3} - 3 x + 1; x + \frac{1}{3}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Using remainder theorem.

Step 3. Using factor theorem.

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Use the Remainder Theorem to find the remainder when f(x)is divided by x-c. Then use the Factor Theorem to determine whether x-cis a factor of f(x).f(x)=3x4+x3-3x+1;x+13

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Using remainder theorem.

Step 3. Using factor theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

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Study anywhere. Anytime. Across all devices.

Use the Remainder Theorem to find the remainder when $f (x)$ is divided by $x - c$ . Then use the Factor Theorem to determine whether $x - c$ is a factor of $f (x)$ .
$f (x) = 3 x^{4} + x^{3} - 3 x + 1; x + \frac{1}{3}$