Question

divide. if the polynomial does not divide evenly, include the remainder as a fraction.\\((-12s^{3} - 56s^{2} + 23s + 61) \div (s + 5)\\)

Explanation:

Step1: Use polynomial long division

Divide the leading term of the dividend \(-12s^3\) by the leading term of the divisor \(s\) to get \(-12s^2\). Multiply the divisor \((s + 5)\) by \(-12s^2\) to get \(-12s^3 - 60s^2\). Subtract this from the dividend:
\[

$$\begin{align*} (-12s^3 - 56s^2 + 23s + 61) - (-12s^3 - 60s^2) &= (-12s^3 - 56s^2 + 23s + 61) + 12s^3 + 60s^2\\ &= 4s^2 + 23s + 61 \end{align*}$$

\]

Step2: Divide the new leading term

Divide the leading term of the new dividend \(4s^2\) by the leading term of the divisor \(s\) to get \(4s\). Multiply the divisor \((s + 5)\) by \(4s\) to get \(4s^2 + 20s\). Subtract this from the new dividend:
\[

$$\begin{align*} (4s^2 + 23s + 61) - (4s^2 + 20s) &= 4s^2 + 23s + 61 - 4s^2 - 20s\\ &= 3s + 61 \end{align*}$$

\]

Step3: Divide the new leading term again

Divide the leading term of the new dividend \(3s\) by the leading term of the divisor \(s\) to get \(3\). Multiply the divisor \((s + 5)\) by \(3\) to get \(3s + 15\). Subtract this from the new dividend:
\[

$$\begin{align*} (3s + 61) - (3s + 15) &= 3s + 61 - 3s - 15\\ &= 46 \end{align*}$$

\]

Step4: Combine the results

The quotient is \(-12s^2 + 4s + 3\) and the remainder is \(46\). So the result of the division is:
\[
-12s^2 + 4s + 3 + \frac{46}{s + 5}
\]

Answer:

\(-12s^2 + 4s + 3 + \frac{46}{s + 5}\)