Question

Divide the polynomial n^3 + 7n^2 + 14n + 3 by (n + 2) to determine the quotient and the remainder. (1 point)

A. The quotient is n^2 - 3n + 11 and the remainder is 6.
B. The quotient is n^2 - 3n + 11 and the remainder is -5.
C. The quotient is n^2 + 5n + 4 and the remainder is 6.
D. The quotient is n^2 + 5n + 4 and the remainder is -5

Answer 1

To divide the polynomial \( n^3 + 7n^2 + 14n + 3 \) by \( n + 2 \), we can use polynomial long division.

1. **Divide the leading term:** Divide the leading term of the dividend \( n^3 \) by the leading term of the divisor \( n \):
\[
\frac{n^3}{n} = n^2
\]

2. **Multiply and subtract:** Multiply \( n^2 \) by the entire divisor \( n + 2 \) and subtract from the original polynomial:
\[
n^3 + 7n^2 + 14n + 3 - (n^2(n + 2)) = n^3 + 7n^2 + 14n + 3 - (n^3 + 2n^2) = 5n^2 + 14n + 3
\]

3. **Repeat the process:** Now take \( 5n^2 \) and divide by \( n \):
\[
\frac{5n^2}{n} = 5n
\]

Multiply \( 5n \) by \( n + 2 \):
\[
5n(n + 2) = 5n^2 + 10n
\]
Subtract this from \( 5n^2 + 14n + 3 \):
\[
5n^2 + 14n + 3 - (5n^2 + 10n) = 4n + 3
\]

4. **Repeat again:** Now take \( 4n \) and divide by \( n \):
\[
\frac{4n}{n} = 4
\]

Multiply 4 by \( n + 2 \):
\[
4(n + 2) = 4n + 8
\]
Subtract this from \( 4n + 3 \):
\[
4n + 3 - (4n + 8) = 3 - 8 = -5
\]

5. **Conclusion:** We have finished dividing. The quotient is:
\[
n^2 + 5n + 4
\]
and the remainder is:
\[
-5
\]

Thus, the answer is:
**D. The quotient is \( n^2 + 5n + 4 \) and the remainder is -5.**