Question

Divide the following polynomials and express your answer as a rational expression:

(x^3 + 2x^2 + 8x + 1) ÷ (x + 5).

Answer 1

The division of the polynomial (x^3 + 2x^2 + 8x + 1) by (x + 5) is performed using long division, resulting in a quotient that is a rational expression and confirming the answer by multiplication and addition of any remainder.

Step-by-step explanation:

To divide the polynomial (x^3 + 2x^2 + 8x + 1) by (x + 5), we use long division or synthetic division. We'll proceed with long division as follows:

• Divide the first term of the numerator by the first term of the denominator: x^3 ÷ x = x^2.
• Multiply the divisor (x + 5) by x^2 and subtract the result from the numerator.
• Bring down the next term of the numerator and repeat these steps until all terms are accounted for.
• If the degree of the remaining polynomial is less than the degree of the divisor, that will be the remainder.

After performing the division, the quotient will be the result, expressed as a rational expression, with the remainder, if there is one, written over the divisor.

To check the answer, multiply the quotient and the divisor together and add the remainder, if any, which should give you the original numerator.