Chapter 15, Problem 1RE

To determine

Whether the following statement is true or false “If $3x^2+5x-8$ is divided by $x-2$, the remainder is 12”.

Answer to Problem 1RE

The statement “If $3x^2+5x-8$ is divided by $x-2$, the remainder is 12” is false statement.

Explanation of Solution

Procedure used:

Procedure for Synthetic Division:

“1. Write the coefficients of $f\left(x\right)$. Be certain that the powers are in descending order and that zeros are inserted for missing powers.

2. Carry down the left coefficient, then multiply it by $r$, and place this product under the second coefficient of the top line.

3. Add the two numbers in the second column and place the result below. Multiply this sum by $r$ and place the product under the third coefficient of the top line.

4. Continue this process until the bottom row has as many numbers as the top row”.

Calculation:

Divide the polynomial $3x^2+5x-8$ by $x-2$ as follows.

The powers of $x$ are in descending order, write down the coefficient of $f\left(x\right)$ and zeros are inserted for missing powers, write $2$ for the divisor $x-2$ and place $2$ to the right.

$35-8\overline{)2}$

Carry the left coefficient 3 to the bottom line and multiply it by $2$, and place the product 6 under the second coefficient $5$ of the top line.

Proceed in this similar manner, to find the quotient and remainder.

$\begin{array}{l}35-8\overline{)2}\\ \underset{\_}{622}\\ 31114\end{array}$

The numbers presented in the last row denotes the coefficients of the quotient and the remainder.

Thus, the remainder is $14$.

Hence, the statement “If $3x^2+5x-8$ is divided by $x-2$, the remainder is 12” is false statement.