Question

Divide the polynomials. your answer should be in the form p(x) k/x 5 where p is a polynomial and k is an integer. (x² - 2x-8)/(x 1)

Option 1: (x - 3) 5
Option 2: (x + 3) 5
Option 3: (x - 4) 5
Option 4: (x + 4) 5

Answer 1

Using polynomial long division, the polynomials (x² - 2x - 8)/(x + 1) divide to give a quotient of (x - 3) with a remainder of -5. Thus, the final expression is (x - 3) 5/x + 5.

Step-by-step explanation:

The student was asked to divide the polynomials x² - 2x - 8 by x + 1 and express the answer in the form p(x) k/x + 5 where p is a polynomial and k is an integer.

To solve this, we use polynomial long division:

1. Divide the first term of the numerator by the first term of the denominator: x² by x, which gives us x.
2. Multiply x by the entire denominator x + 1 giving us x² + x.
3. Subtract this from the original polynomial: (x² - 2x) - (x² + x) which simplifies to -3x - 8.
4. Repeat the process now using -3x divided by x which gives -3.
5. Multiply -3 by x + 1 to get -3x - 3 and subtract this from -3x - 8 resulting in -5.

This gives us a quotient of x - 3 with a remainder of -5. The final answer, in the requested format, is (x - 3) 5/x + 5, which corresponds to Option 1.