Question

Use synthetic division and the given factor to completely factor the polynomial function.

x³ + 8x² + 4x-48; x+6

Answer 1

To factor the polynomial function x³ + 8x² + 4x - 48 using synthetic division with the factor x + 6, we perform the synthetic division to obtain the quotient polynomial. Then we factor the quotient to find the remaining factors, which combined with the given factor give us the fully factored form of the initial polynomial: (x + 6)(x + 4)(x - 2).

Step-by-step explanation:

To completely factor the polynomial function x³ + 8x² + 4x - 48 using synthetic division with the given factor x + 6, follow these steps:

1. Set up synthetic division by writing the coefficients of the polynomial in descending order and placing the zero of the given factor, which is -6, to the left.
2. Bring down the leading coefficient, which is 1, to the bottom row.
3. Multiply the bottom left number (-6) by the number just brought down (1) and write the product (-6) above the next coefficient (8), under the horizontal line.
4. Add the second coefficient and the product to get the new second number in the bottom row.
5. Continue this process of multiplying and adding across all coefficients.
6. Once completed, the bottom row gives the coefficients of the quotient polynomial.
7. Since the given factor is x + 6, the remaining polynomial will be one degree less than the original, thus x² + 2x - 8.
8. Factor the quotient polynomial, which can be factored to (x + 4)(x - 2).
9. Combine the given factor with the factored quotient to express the completely factored form of the original polynomial: (x + 6)(x + 4)(x - 2).

Now, the polynomial x³ + 8x² + 4x - 48 is completely factored using synthetic division and can be expressed as the product of three factors.