Question

1. $(20c^{3}+22c^{2}-7c + 7)div(5c - 2)$

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(20c^{3}\) by the leading term of the divisor \(5c\). So, \(\frac{20c^{3}}{5c} = 4c^{2}\).

Step2: Multiply and subtract

Multiply the divisor \(5c - 2\) by \(4c^{2}\) to get \(20c^{3}-8c^{2}\). Subtract this from the dividend:
\[

$$\begin{align*} &(20c^{3}+22c^{2}-7c + 7)-(20c^{3}-8c^{2})\\ =&20c^{3}+22c^{2}-7c + 7 - 20c^{3}+8c^{2}\\ =&30c^{2}-7c + 7 \end{align*}$$

\]

Step3: Divide the new leading term

Divide the leading term of the new dividend \(30c^{2}\) by the leading term of the divisor \(5c\). So, \(\frac{30c^{2}}{5c}=6c\).

Step4: Multiply and subtract

Multiply the divisor \(5c - 2\) by \(6c\) to get \(30c^{2}-12c\). Subtract this from the new dividend:
\[

$$\begin{align*} &(30c^{2}-7c + 7)-(30c^{2}-12c)\\ =&30c^{2}-7c + 7 - 30c^{2}+12c\\ =&5c + 7 \end{align*}$$

\]

Step5: Divide the new leading term

Divide the leading term of the new dividend \(5c\) by the leading term of the divisor \(5c\). So, \(\frac{5c}{5c} = 1\).

Step6: Multiply and subtract

Multiply the divisor \(5c - 2\) by \(1\) to get \(5c - 2\). Subtract this from the new dividend:
\[

$$\begin{align*} &(5c + 7)-(5c - 2)\\ =&5c + 7 - 5c + 2\\ =&9 \end{align*}$$

\]

Answer:

The quotient is \(4c^{2}+6c + 1\) and the remainder is \(9\), so \((20c^{3}+22c^{2}-7c + 7)\div(5c - 2)=4c^{2}+6c + 1+\frac{9}{5c - 2}\)