Question

Find (x³ + 7x² - 16x - 112/x + 4) using synthetic division.

(A) Perform synthetic division to find the quotient
(B) Synthetic division is not applicable
(C) The result cannot be determined
(D) None of the above

Answer 1

Synthetic division can be used to divide the polynomial (x³ + 7x² - 16x - 112) by (x + 4), yielding a quotient of x² + 3x - 4 with a remainder of 0. The correct choice is (A).

Step-by-step explanation:

To perform synthetic division of the given polynomial (x³ + 7x² - 16x - 112)/(x + 4), we use -4 as the divisor (since we are dividing by x + 4, we take the opposite sign of the constant in the divisor). The steps for synthetic division are as follows:

1. Write down the coefficients of the polynomial: 1, 7, -16, -112.
2. Bring down the leading coefficient to the bottom row.
3. Multiply this coefficient by -4 and write the result under the next coefficient.
4. Add the numbers in the second column to get the new number at the bottom.
5. Repeat this process until all coefficients have been used.
6. The final bottom row, excluding the last number, represents the coefficients of the quotient polynomial.

The synthetic division would proceed as:

• -4 | 1 7 -16 -112
• | -4 12 16
• 1 3 -4 0

Thus, the quotient is x² + 3x - 4 and the remainder is 0 since the last number in the bottom row is 0. The correct answer is (A).