Question

Divide (x^5 + 6x^4 - 3x^2 - 22x - 29) by (x + 6).

a) x^4 - 6x^3 + 33x^2 + 195x + 1166
b) x^4 - 6x^3 + 33x^2 + 195x - 325
c) x^4 + 6x^3 + 33x^2 - 195x - 29
d) x^4 + 6x^3 + 33x^2 - 195x + 29

Answer 1

To divide the given polynomial by (x + 6), synthetic division is used. The process involves writing down coefficients, using the zero of the divisor, and performing multiplications and additions to find the quotient. The correct result is (a) x^4 - 6x^3 + 33x^2 + 195x + 1166.

Step-by-step explanation:

To divide the polynomial (x^5 + 6x^4 - 3x^2 - 22x - 29) by (x + 6), we can use polynomial long division or synthetic division. Since the divisor is a first-degree polynomial, synthetic division is a faster approach.

1. We write down the coefficients of the dividend: 1, 6, 0, -3, -22, -29. (Notice that there is a 0 for the x^3 term which is not present in the polynomial).
2. Write down the zero of the divisor, which is -6 (because x + 6 = 0 when x = -6).
3. Bring down the leading coefficient (1) to the bottom row.
4. Multiply this by -6 and write the result under the next coefficient.
5. Add the numbers in the second column and write the result in the bottom row.
6. Repeat this process for each column.

The result is the coefficients of the quotient polynomial, which corresponds to option (a) x^4 - 6x^3 + 33x^2 + 195x + 1166.