Question

question: use the long - division method to find the result when 3x^3 - 7x^2 - 6x + 16 is divided by 3x - 7. if there is a remainder, express the result in the form q(x)+\frac{r}{b(x)}.

Explanation:

Step1: Set up long - division

We are dividing the polynomial \(3x^{3}-7x^{2}-6x + 16\) by \(3x - 7\). First, divide the leading term of the dividend \(3x^{3}\) by the leading term of the divisor \(3x\). \(\frac{3x^{3}}{3x}=x^{2}\).

Step2: Multiply and subtract

Multiply \(3x - 7\) by \(x^{2}\) to get \(3x^{3}-7x^{2}\), and subtract this from the dividend: \((3x^{3}-7x^{2}-6x + 16)-(3x^{3}-7x^{2})=-6x + 16\).

Step3: Divide the new leading term

Divide the leading term of \(-6x+16\) (which is \(-6x\)) by the leading term of the divisor \(3x\). \(\frac{-6x}{3x}=-2\).

Step4: Multiply and subtract again

Multiply \(3x - 7\) by \(-2\) to get \(-6x + 14\), and subtract this from \(-6x + 16\): \((-6x + 16)-(-6x + 14)=2\).

Answer:

\(x^{2}-2+\frac{2}{3x - 7}\)