Chapter C, Problem 2YT

To determine

To calculate: The value of $\left(3x-9+2x^3\right)÷\left(x+2\right)$ using synthetic division.

Answer to Problem 2YT

Solution:

The value of $\left(3x-9+2x^3\right)÷\left(x+2\right)$ is $\underset{\_}{2x^2-4x+11+\frac{\left(-31\right)}{x+2}}$.

Explanation of Solution

Given Information:

The expression is $\left(3x-9+2x^3\right)÷\left(x+2\right)$.

Formula used:

The synthetic division is explained as:

The divisor must be in the form of $x-a$ which is a linear expression. Where, $x$ is a variable and $a$ is constant. The coefficient of the variable must be $1$.

Calculation:

Consider the expression:

$\left(3x-9+2x^3\right)÷\left(x+2\right)$

Rewrite the dividend $3x-9+2x^3$ in descending order as $2x^3+3x-9$ and the divisor $\left(x+2\right)$ as $\left(x-\left(-2\right)\right)$.

Since, the dividend has no $x^2$ term. So, write 0 as the coefficient of $x^2$.

Thus, the dividend is $2x^3+0x^2+3x-9$.

The constant term of the divisor 2 with the opposite sign is written to left.

The coefficients of the dividend 2, 0, 3, $-9$ are written to the right.

Evaluate the value of $\left(3x-9+2x^3\right)÷\left(x-\left(-2\right)\right)$ using synthetic division as follows.

$\begin{array}{l}\overline{)-2}\text{ 2 0 3 }-9\\ \underset{\_}{-4\text{ 8 }-22}\\ \text{ 2 }-4\text{ 11 }-31\end{array}$

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

So, the polynomial with coefficients $2,-4,11$ is $2x^2-4x+11$ and the remainder is $-31$.

Therefore, the value of $\left(3x-9+2x^3\right)÷\left(x+2\right)$ is $2x^2-4x+11+\frac{\left(-31\right)}{x+2}$.