use the long division method to find the result when $3x^4 - 30x^3 + 11x^2 + 10x - 4$ is divided by $3x^2 - 1$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Question

Accepted Answer

# Explanation:
## Step1: Divide the leading terms
Divide the leading term of the dividend $3x^4$ by the leading term of the divisor $3x^2$. We get $\frac{3x^4}{3x^2}=x^2$. Multiply the divisor $3x^2 - 1$ by $x^2$ to get $3x^4 - x^2$. Subtract this from the dividend:
$$
\begin{align*}
&(3x^4 - 30x^3 + 11x^2 + 10x - 4)-(3x^4 - x^2)\
=&3x^4 - 30x^3 + 11x^2 + 10x - 4 - 3x^4 + x^2\
=& - 30x^3 + 12x^2 + 10x - 4
\end{align*}
$$
## Step2: Divide the new leading terms
Divide the leading term of the new dividend $-30x^3$ by the leading term of the divisor $3x^2$. We get $\frac{-30x^3}{3x^2}=-10x$. Multiply the divisor $3x^2 - 1$ by $-10x$ to get $-30x^3 + 10x$. Subtract this from the new dividend:
$$
\begin{align*}
&(-30x^3 + 12x^2 + 10x - 4)-(-30x^3 + 10x)\
=&-30x^3 + 12x^2 + 10x - 4 + 30x^3 - 10x\
=&12x^2 - 4
\end{align*}
$$
## Step3: Divide the new leading terms again
Divide the leading term of the new dividend $12x^2$ by the leading term of the divisor $3x^2$. We get $\frac{12x^2}{3x^2} = 4$. Multiply the divisor $3x^2 - 1$ by $4$ to get $12x^2 - 4$. Subtract this from the new dividend:
$$
\begin{align*}
&(12x^2 - 4)-(12x^2 - 4)\
=&12x^2 - 4 - 12x^2 + 4\
=&0
\end{align*}
$$
The quotient $q(x)$ is the sum of the terms we found in each step: $x^2-10x + 4$, and the remainder $r(x)=0$.

# Answer:
$x^{2}-10x + 4$ (since the remainder is $0$, the form $q(x)+\frac{r(x)}{b(x)}$ simplifies to just $q(x)$ as $\frac{0}{3x^{2}-1}=0$)

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QUESTION IMAGE

Question

Explanation:

Step1: Divide the leading terms

Step2: Divide the new leading terms

Step3: Divide the new leading terms again

Answer: