Question

using long division determine which of the following polynomials divides evenly for a divisor of x+3
a: 3x^2-6x+9
b: 5x^2 +7x -12
c: x^3-3x^2 -13x+15

Answer 1

To determine which of the given polynomials divides evenly by \( x + 3 \), we can use synthetic division or long division. Since you're interested in using long division, I'll demonstrate using long division for each polynomial.

### a: \( 3x^2 - 6x + 9 \)

1. Divide the leading term: \( \frac{3x^2}{x} = 3x \).
2. Multiply \( 3x \) by \( x + 3 \):
- \( 3x(x + 3) = 3x^2 + 9x \).
3. Subtract this from the original polynomial:
\[
(3x^2 - 6x + 9) - (3x^2 + 9x) = -15x + 9.
\]
4. Now divide the leading term: \( \frac{-15x}{x} = -15 \).
5. Multiply \( -15 \) by \( x + 3 \):
- \( -15(x + 3) = -15x - 45 \).
6. Subtract:
\[
(-15x + 9) - (-15x - 45) = 54.
\]

The remainder is \( 54 \), so \( 3x^2 - 6x + 9 \) does not divide evenly by \( x + 3 \).

### b: \( 5x^2 + 7x - 12 \)

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

The remainder is \( 12 \), so \( 5x^2 + 7x - 12 \) does not divide evenly by \( x + 3 \).

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

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

The remainder is \( 0 \), so \( x^3 - 3x^2 - 13x + 15 \) divides evenly by \( x + 3 \).

### Conclusion
Only polynomial **c** \( x^3 - 3x^2 - 13x + 15 \) divides evenly by \( x + 3 \).