Question

what is the quotient of ((x^4 + 8x^3 - 3x^2 - 24x) div (x - 3))?

a) (x^3 - 11x^2 + 30x + 66 + \frac{198}{x - 3})

b) (x^3 + 11x^2 - 30x - 66 + \frac{198}{x - 3})

c) (x^3 - 11x^2 - 30x - 66 + \frac{198}{x - 3})

d) (x^3 + 11x^2 + 30x + 66 + \frac{198}{x - 3})

Explanation:

Step1: Use polynomial long division

Divide \(x^4 + 8x^3 - 3x^2 - 24x\) by \(x - 3\).
First term: \(\frac{x^4}{x}=x^3\). Multiply \(x - 3\) by \(x^3\): \(x^4 - 3x^3\). Subtract from the original polynomial: \((x^4 + 8x^3 - 3x^2 - 24x)-(x^4 - 3x^3)=11x^3 - 3x^2 - 24x\).

Step2: Next term

\(\frac{11x^3}{x}=11x^2\). Multiply \(x - 3\) by \(11x^2\): \(11x^3 - 33x^2\). Subtract: \((11x^3 - 3x^2 - 24x)-(11x^3 - 33x^2)=30x^2 - 24x\).

Step3: Next term

\(\frac{30x^2}{x}=30x\). Multiply \(x - 3\) by \(30x\): \(30x^2 - 90x\). Subtract: \((30x^2 - 24x)-(30x^2 - 90x)=66x\).

Step4: Next term

\(\frac{66x}{x}=66\). Multiply \(x - 3\) by \(66\): \(66x - 198\). Subtract: \((66x)-(66x - 198)=198\).

So the quotient is \(x^3 + 11x^2 + 30x + 66\) with a remainder of \(198\), so the expression is \(x^3 + 11x^2 + 30x + 66 + \frac{198}{x - 3}\).

Answer:

D. \(x^3 + 11x^2 + 30x + 66 + \frac{198}{x - 3}\)