Question

find the quotient and remainder using long or synthetic division for \\(\frac{x^2 - 3x - 3}{x - 5}\\). give your answer in the form: \\(\text{quotient} + \frac{\text{remainder}}{\text{divisor}}\\)

Explanation:

Step1: Set up long division

Divide \(x^2 - 3x - 3\) by \(x - 5\). First, divide the leading term \(x^2\) by \(x\) to get \(x\).

Step2: Multiply and subtract

Multiply \(x\) by \(x - 5\) to get \(x^2 - 5x\). Subtract this from \(x^2 - 3x - 3\): \((x^2 - 3x - 3)-(x^2 - 5x)=2x - 3\).

Step3: Divide the new leading term

Divide \(2x\) by \(x\) to get \(2\).

Step4: Multiply and subtract again

Multiply \(2\) by \(x - 5\) to get \(2x - 10\). Subtract this from \(2x - 3\): \((2x - 3)-(2x - 10)=7\).

Answer:

\(x + 2+\frac{7}{x - 5}\)