Question

use division to express the fraction in the form qu\
\\(\frac{x^2 - x - 7}{x - 2}\\)\
\\(\frac{x^2 - x - 7}{x - 2} = \square\\)\
(simplify your answer.)

Explanation:

Step1: Perform Polynomial Long Division

Divide \(x^2 - x - 7\) by \(x - 2\). First, divide the leading term \(x^2\) by \(x\) to get \(x\). Multiply \(x - 2\) by \(x\) to get \(x^2 - 2x\). Subtract this from \(x^2 - x - 7\): \((x^2 - x - 7) - (x^2 - 2x) = x - 7\).

Step2: Divide the New Leading Term

Now, divide the leading term \(x\) of \(x - 7\) by \(x\) to get \(1\). Multiply \(x - 2\) by \(1\) to get \(x - 2\). Subtract this from \(x - 7\): \((x - 7) - (x - 2) = -5\).

Step3: Write the Result

The division gives a quotient of \(x + 1\) and a remainder of \(-5\). So, \(\frac{x^2 - x - 7}{x - 2}=x + 1+\frac{-5}{x - 2}\) or \(x + 1-\frac{5}{x - 2}\).

Answer:

\(x + 1 - \dfrac{5}{x - 2}\)