Chapter 2.3, Problem 46E

a .

To determine

To find: $f\left(1\right)$ using Remainder Theorem and synthetic division.

Answer to Problem 46E

  $f\left(1\right)=19$

Explanation of Solution

Given:

The function $f\left(x\right)=4x^4-12x^3+11x^2+16$

Calculation:

Remainder theorem:

If on dividing a polynomial $P\left(x\right)$ by $x-r$ , the remainder obtained is R . Then,

  $P\left(r\right)=R$

The given expression can be written as:

  $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$

Thus, using this theorem $f\left(1\right)$ is the remainder R obtained by dividing $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$ by $x-1$

Now, to find this remainder R use synthetic division.

Then, use synthetic division to divide $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$ by $x-1$ .

  $\begin{array}{l}\text{ 4 }-12\text{ 11 0 16}\\ \text{ 4 }-83\text{ 3}\\ 1\overline{)\text{4 }-833\text{ 19 }}=R=f\left(1\right)\end{array}$

Hence, $f\left(1\right)=19$

b .

To determine

To find: $f\left(-2\right)$ using Remainder Theorem and synthetic division.

Answer to Problem 46E

  $f\left(-2\right)=220$

Explanation of Solution

Given:

The function $f\left(x\right)=4x^4-12x^3+11x^2+16$

Calculation:

Remainder theorem:

If on dividing a polynomial $P\left(x\right)$ by $x-r$ , the remainder obtained is R . Then,

  $P\left(r\right)=R$

The given expression can be written as:

  $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$

Thus, using this theorem $f\left(-2\right)$ is the remainder R obtained by dividing $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$ by $x+2$

Now, to find this remainder R use synthetic division.

Then, use synthetic division to divide $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$ by $x+2$ .

  $\begin{array}{l}\text{ 4 }-12\text{ 11 0 16}\\ -8\text{ 40 }-\text{102 204}\\ -2\overline{)\text{4 }-20\text{ 51 }-102\text{ 220 }}=R=f\left(-2\right)\end{array}$

Hence, $f\left(-2\right)=220$

c .

To determine

To find: $f\left(3\right)$ using Remainder Theorem and synthetic division.

Answer to Problem 46E

  $f\left(3\right)=115$

Explanation of Solution

Given:

The function $f\left(x\right)=4x^4-12x^3+11x^2+16$

Calculation:

Remainder theorem:

If on dividing a polynomial $P\left(x\right)$ by $x-r$ , the remainder obtained is R . Then,

  $P\left(r\right)=R$

The given expression can be written as:

  $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$

Thus, using this theorem $f\left(3\right)$ is the remainder R obtained by dividing $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$ by $x-3$

Now, to find this remainder R use synthetic division.

Then, use synthetic division to divide $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$ by $x-3$ .

  $\begin{array}{l}\text{ 4 }-12\text{ 11 0 16}\\ \text{ 12 0 }33\text{ 99}\\ 3\overline{)\text{4 0 11 }33\text{ 115 }}=R=f\left(3\right)\end{array}$

Hence, $f\left(3\right)=115$

d .

To determine

To find: $f\left(-6\right)$ using Remainder Theorem and synthetic division.

Answer to Problem 46E

  $f\left(-6\right)=8188$

Explanation of Solution

Given:

The function $f\left(x\right)=4x^4-12x^3+11x^2+16$

Calculation:

Remainder theorem:

If on dividing a polynomial $P\left(x\right)$ by $x-r$ , the remainder obtained is R . Then,

  $P\left(r\right)=R$

The given expression can be written as:

  $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$

Thus, using this theorem $f\left(-6\right)$ is the remainder R obtained by dividing $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$ by $x+6$

Now, to find this remainder R use synthetic division.

Then, use synthetic division to divide $f\left(x\right)=4x^4-12x^3+11x^2+0x+16$ by $x+6$ .

  $\begin{array}{l}\text{ 4 }-12\text{ 11 0 16}\\ -24\text{ 216 }-1362\text{ 8172}\\ -6\overline{)\text{4 }-36\text{ 227 }-1362\text{ 8188 }}=R=f\left(-6\right)\end{array}$

Hence, $f\left(-6\right)=8188$