Use the image to answer the question.

Question

x squared minus 4 is divided by x minus 1 using the long division method. The first expression of divisibility, left parenthesis x squared minus x right parenthesis, is subtracted from the dividend. The remainder is x minus 4. The quotient is x.

The image shows the first few steps of the polynomial division (x2−4)÷(x−1)
. Complete the polynomial division process, then give the quotient and the remainder expressed as a fraction.

(1 point)

Answer 1

To complete the polynomial division of \((x^2 - 4) \div (x - 1)\) using long division, follow these steps:

1. Divide the leading term: Divide \(x^2\) (the leading term of the dividend) by \(x\) (the leading term of the divisor) to get \(x\).

2. Multiply and subtract: Multiply \(x\) by the entire divisor \((x - 1)\): \[ x \cdot (x - 1) = x^2 - x \] Now, subtract this from the original polynomial: \[ (x^2 - 4) - (x^2 - x) = -4 + x = x - 4 \]

3. Bring down the next term (if applicable): Here, there are no more terms to bring down.

4. Repeat the process: Now, take \(x - 4\) and divide the leading term \(x\) by \(x\) to get 1. Multiply the entire divisor by 1: \[ 1 \cdot (x - 1) = x - 1 \] Subtract this from \(x - 4\): \[ (x - 4) - (x - 1) = -4 + 1 = -3 \]

5. Final result: Now, we cannot divide anymore since the degree of the remainder (-3) is lower than the degree of the divisor (1).

Thus, the quotient is \(x + 1\) and the remainder is \(-3\).

Final answer:

The quotient is \(x + 1\) and the remainder is \(-3\).

If we express this as a fraction: \[ \frac{(x^2 - 4)}{(x - 1)} = x + 1 - \frac{3}{(x - 1)} \]

So the final result is: Quotient: \(x + 1\), Remainder: -3 (or as a fraction: \( \frac{-3}{x - 1} \)).