Chapter C, Problem 1YT

To determine

To calculate: The value of $\left(x^3+2x^2-17x+6\right)÷\left(x-3\right)$ using synthetic division.

Answer to Problem 1YT

Solution:

The value of $\left(x^3+2x^2-17x+6\right)÷\left(x-3\right)$ is $\underset{\_}{x^2+5x-2}$.

Explanation of Solution

Given Information:

The expression is $\left(x^3+2x^2-17x+6\right)÷\left(x-3\right)$.

Calculation:

The constant term of the divisor is $-3$ with the opposite sign is written to left.

The coefficients of the dividend 1, 2, $-17$, 6 are written to the right.

Evaluate the value of $\left(x^3+2x^2-17x+6\right)÷\left(x-3\right)$ using synthetic division as follows.

$\begin{array}{l}\overline{)3}\text{ 1 2 }-\text{17 6}\\ \underset{\_}{}\\ \text{ 1}\\ \end{array}$

Multiply 1 by 3 and add 2 to the result.

$\begin{array}{l}\overline{)3}\text{ 1 2 }-\text{17 6}\\ \underset{\_}{\text{ 3 }}\\ \text{ 1 5}\end{array}$

Multiply 3 by 5 and add $-17$ to the result.

$\begin{array}{l}\overline{)3}\text{ 1 2 }-\text{17 6}\\ \underset{\_}{\text{ 3 15 }}\\ \text{ 1 5 }-2\\ \end{array}$

Multiply 3 by $-2$ and add 6 to the result.

$\begin{array}{l}\overline{)3}\text{ 1 2 }-\text{17 6}\\ \underset{\_}{\text{ 3 15 }-6}\\ \text{ 1 5 }-2\text{ 0}\\ \end{array}$

Finally, replace the variables in descending order. The maximum degree of the dividend is $3$. Thus, the maximum degree of the quotient should be $2$.

$\begin{array}{l}\overline{)3}\text{ 1 2 }-\text{17 6}\\ \underset{\_}{\text{ 3 15 }-6}\\ \text{ 1}{x}^2\text{ 5}x-2\text{ 0}\\ \end{array}$

The obtained numbers are the coefficients of a polynomial and the last digit is the remainder.

So, the polynomial with coefficients 1, 5, $-2$ is $x^2+5x-2$.

Therefore, the value of $\left(x^3+2x^2-17x+6\right)÷\left(x-3\right)$ is $x^2+5x-2$.