Question

divide. if the polynomial does not divide evenly, include the remainder as a fraction.
(-s³ + 105s - 50) ÷ (s - 10)

Explanation:

Step1: Use polynomial long division

We divide \(-s^3 + 105s - 50\) by \(s - 10\). First, divide the leading term \(-s^3\) by \(s\) to get \(-s^2\). Multiply \(s - 10\) by \(-s^2\) to get \(-s^3 + 10s^2\). Subtract this from the original polynomial:
\[

$$\begin{align*} (-s^3 + 105s - 50) - (-s^3 + 10s^2) &= -s^3 + 105s - 50 + s^3 - 10s^2\\ &= -10s^2 + 105s - 50 \end{align*}$$

\]

Step2: Divide the new leading term

Divide \(-10s^2\) by \(s\) to get \(-10s\). Multiply \(s - 10\) by \(-10s\) to get \(-10s^2 + 100s\). Subtract this from the previous result:
\[

$$\begin{align*} (-10s^2 + 105s - 50) - (-10s^2 + 100s) &= -10s^2 + 105s - 50 + 10s^2 - 100s\\ &= 5s - 50 \end{align*}$$

\]

Step3: Divide the new leading term

Divide \(5s\) by \(s\) to get \(5\). Multiply \(s - 10\) by \(5\) to get \(5s - 50\). Subtract this from the previous result:
\[

$$\begin{align*} (5s - 50) - (5s - 50) &= 5s - 50 - 5s + 50\\ &= 0 \end{align*}$$

\]

Step4: Combine the results

The quotient is \(-s^2 - 10s + 5\) and the remainder is \(0\).

Answer:

\(-s^2 - 10s + 5\)