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: \( 3x^2 - 6x + 9 \)
- Divide the leading term: \( \frac{3x^2}{x} = 3x \).
- Multiply \( 3x \) by \( x + 3 \):
- \( 3x(x + 3) = 3x^2 + 9x \).
- Subtract this from the original polynomial: \[ (3x^2 - 6x + 9) - (3x^2 + 9x) = -15x + 9. \]
- Now divide the leading term: \( \frac{-15x}{x} = -15 \).
- Multiply \( -15 \) by \( x + 3 \):
- \( -15(x + 3) = -15x - 45 \).
- Subtract: \[ (-15x + 9) - (-15x - 45) = 54. \]
The remainder is \( 54 \), so \( 3x^2 - 6x + 9 \) does not divide evenly by \( x + 3 \).
b: \( 5x^2 + 7x - 12 \)
- Divide the leading term: \( \frac{5x^2}{x} = 5x \).
- Multiply \( 5x \) by \( x + 3 \):
- \( 5x(x + 3) = 5x^2 + 15x \).
- Subtract: \[ (5x^2 + 7x - 12) - (5x^2 + 15x) = -8x - 12. \]
- Now divide the leading term: \( \frac{-8x}{x} = -8 \).
- Multiply \( -8 \) by \( x + 3 \):
- \( -8(x + 3) = -8x - 24 \).
- Subtract: \[ (-8x - 12) - (-8x - 24) = 12. \]
The remainder is \( 12 \), so \( 5x^2 + 7x - 12 \) does not divide evenly by \( x + 3 \).
c: \( x^3 - 3x^2 - 13x + 15 \)
- Divide the leading term: \( \frac{x^3}{x} = x^2 \).
- Multiply \( x^2 \) by \( x + 3 \):
- \( x^2(x + 3) = x^3 + 3x^2 \).
- Subtract: \[ (x^3 - 3x^2 - 13x + 15) - (x^3 + 3x^2) = -6x^2 - 13x + 15. \]
- Now divide the leading term: \( \frac{-6x^2}{x} = -6x \).
- Multiply \( -6x \) by \( x + 3 \):
- \( -6x(x + 3) = -6x^2 - 18x \).
- Subtract: \[ (-6x^2 - 13x + 15) - (-6x^2 - 18x) = 5x + 15. \]
- Now divide the leading term: \( \frac{5x}{x} = 5 \).
- Multiply \( 5 \) by \( x + 3 \):
- \( 5(x + 3) = 5x + 15 \).
- 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 \( x + 3 \).
1 answer