Question

divide. if the polynomial does not divide evenly, include the remainder as a fraction.\\((4h^{3} - 17h^{2} + 10h + 15) \div (h - 3)\\)

Explanation:

Step1: Use polynomial long division

Divide the leading term of the dividend \(4h^3 - 17h^2 + 10h + 15\) by the leading term of the divisor \(h - 3\). The leading term of the dividend is \(4h^3\) and the leading term of the divisor is \(h\), so \(4h^3\div h = 4h^2\). Multiply the divisor \(h - 3\) by \(4h^2\) to get \(4h^3 - 12h^2\). Subtract this from the dividend:
\[

$$\begin{align*} &(4h^3 - 17h^2 + 10h + 15)-(4h^3 - 12h^2)\\ =&4h^3 - 17h^2 + 10h + 15 - 4h^3 + 12h^2\\ =& - 5h^2 + 10h + 15 \end{align*}$$

\]

Step2: Divide the new leading term

The new leading term is \(-5h^2\) and the leading term of the divisor is \(h\), so \(-5h^2\div h=-5h\). Multiply the divisor \(h - 3\) by \(-5h\) to get \(-5h^2 + 15h\). Subtract this from the previous result:
\[

$$\begin{align*} &(-5h^2 + 10h + 15)-(-5h^2 + 15h)\\ =&-5h^2 + 10h + 15 + 5h^2 - 15h\\ =& - 5h + 15 \end{align*}$$

\]

Step3: Divide the new leading term again

The new leading term is \(-5h\) and the leading term of the divisor is \(h\), so \(-5h\div h = - 5\). Multiply the divisor \(h - 3\) by \(-5\) to get \(-5h + 15\). Subtract this from the previous result:
\[

$$\begin{align*} &(-5h + 15)-(-5h + 15)\\ =&-5h + 15 + 5h - 15\\ =&0 \end{align*}$$

\]

Answer:

\(4h^2 - 5h - 5\)