Question

use long division, determine which of the following polynomials divides evenly for a divisor of x+3
A. 3x^2-6x+9
B. x^3-3x^2-13x+15
C. x^3-5x^2+10x-15
D. 5x^2+7x-12

Answer 1

To determine which of the given polynomials divides evenly by the divisor \(x + 3\), we will perform polynomial long division for each option. A polynomial divides evenly by \(x + 3\) if the remainder is 0 after the division.

### A. \(3x^2 - 6x + 9\)

1. Divide the leading term: \(3x^2 ÷ x = 3x\).
2. Multiply \(3x\) by \(x + 3\): \(3x^2 + 9x\).
3. Subtract:
\[
(3x^2 - 6x + 9) - (3x^2 + 9x) = -15x + 9.
\]
4. Divide the leading term: \(-15x ÷ x = -15\).
5. Multiply \(-15\) by \(x + 3\): \(-15x - 45\).
6. Subtract:
\[
(-15x + 9) - (-15x - 45) = 54.
\]

**Remainder**: 54 (not divisible).

### B. \(x^3 - 3x^2 - 13x + 15\)

1. Divide the leading term: \(x^3 ÷ x = x^2\).
2. Multiply \(x^2\) by \(x + 3\): \(x^3 + 3x^2\).
3. Subtract:
\[
(x^3 - 3x^2 - 13x + 15) - (x^3 + 3x^2) = -6x^2 - 13x + 15.
\]
4. Divide the leading term: \(-6x^2 ÷ x = -6x\).
5. Multiply \(-6x\) by \(x + 3\): \(-6x^2 - 18x\).
6. Subtract:
\[
(-6x^2 - 13x + 15) - (-6x^2 - 18x) = 5x + 15.
\]
7. Divide the leading term: \(5x ÷ x = 5\).
8. Multiply \(5\) by \(x + 3\): \(5x + 15\).
9. Subtract:
\[
(5x + 15) - (5x + 15) = 0.
\]

**Remainder**: 0 (divisible).

### C. \(x^3 - 5x^2 + 10x - 15\)

1. Divide the leading term: \(x^3 ÷ x = x^2\).
2. Multiply \(x^2\) by \(x + 3\): \(x^3 + 3x^2\).
3. Subtract:
\[
(x^3 - 5x^2 + 10x - 15) - (x^3 + 3x^2) = -8x^2 + 10x - 15.
\]
4. Divide the leading term: \(-8x^2 ÷ x = -8x\).
5. Multiply \(-8x\) by \(x + 3\): \(-8x^2 - 24x\).
6. Subtract:
\[
(-8x^2 + 10x - 15) - (-8x^2 - 24x) = 34x - 15.
\]
7. Divide the leading term: \(34x ÷ x = 34\).
8. Multiply \(34\) by \(x + 3\): \(34x + 102\).
9. Subtract:
\[
(34x - 15) - (34x + 102) = -117.
\]

**Remainder**: -117 (not divisible).

### D. \(5x^2 + 7x - 12\)

1. Divide the leading term: \(5x^2 ÷ x = 5x\).
2. Multiply \(5x\) by \(x + 3\): \(5x^2 + 15x\).
3. Subtract:
\[
(5x^2 + 7x - 12) - (5x^2 + 15x) = -8x - 12.
\]
4. Divide the leading term: \(-8x ÷ x = -8\).
5. Multiply \(-8\) by \(x + 3\): \(-8x - 24\).
6. Subtract:
\[
(-8x - 12) - (-8x - 24) = 12.
\]

**Remainder**: 12 (not divisible).

### Conclusion:
The only polynomial from the options that divides evenly by \(x + 3\) is **B**: \(x^3 - 3x^2 - 13x + 15\).