Question

use the long division method to find the result when $3x^3 + 16x^2 + 10x + 25$ is divided by $x + 5$.
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attempt 1 out of 2
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Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(3x^{3}+16x^{2}+10x + 25\) (which is \(3x^{3}\)) by the leading term of the divisor \(x + 5\) (which is \(x\)). So, \(\frac{3x^{3}}{x}=3x^{2}\).

Step2: Multiply and subtract

Multiply the divisor \(x + 5\) by \(3x^{2}\) to get \(3x^{3}+15x^{2}\). Subtract this from the dividend:
\[

$$\begin{align*} (3x^{3}+16x^{2}+10x + 25)-(3x^{3}+15x^{2})&=3x^{3}+16x^{2}+10x + 25 - 3x^{3}-15x^{2}\\ &=x^{2}+10x + 25 \end{align*}$$

\]

Step3: Divide the new leading terms

Now, divide the leading term of the new dividend \(x^{2}+10x + 25\) (which is \(x^{2}\)) by the leading term of the divisor \(x\). So, \(\frac{x^{2}}{x}=x\).

Step4: Multiply and subtract again

Multiply the divisor \(x + 5\) by \(x\) to get \(x^{2}+5x\). Subtract this from \(x^{2}+10x + 25\):
\[

$$\begin{align*} (x^{2}+10x + 25)-(x^{2}+5x)&=x^{2}+10x + 25 - x^{2}-5x\\ &=5x + 25 \end{align*}$$

\]

Step5: Divide the new leading terms again

Divide the leading term of the new dividend \(5x + 25\) (which is \(5x\)) by the leading term of the divisor \(x\). So, \(\frac{5x}{x}=5\).

Step6: Multiply and subtract the last time

Multiply the divisor \(x + 5\) by \(5\) to get \(5x + 25\). Subtract this from \(5x + 25\):
\[
(5x + 25)-(5x + 25)=0
\]

Step7: Combine the results

The terms we got from each division step are \(3x^{2}\), \(x\), and \(5\). So the result of the division is \(3x^{2}+x + 5\).

Answer:

\(3x^{2}+x + 5\)