use long division, determine which of the following polynomials divides evenly for a divisor of x+3

Question

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