Question

Which of the following is a factor of the polynomial x³+x²-17x+15?

• A. x+3
• B. x-3
• C. 2x+3
• D. 2x-3

Answer 1

The Correct Option is B

Step 1: Recall the Factor Theorem.

The Factor Theorem states that if \( (x - c) \) is a factor of a polynomial \( P(x) \), then \( P(c) = 0 \). To check if any of the given options is a factor, we substitute the corresponding value of \( c \) into \( P(x) \) and verify if \( P(c) = 0 \).

Step 2: Test each option.

(1) \( x + 3 \):

If \( x + 3 \) is a factor, then \( c = -3 \). Substitute \( x = -3 \) into \( P(x) \):

\[ P(-3) = (-3)^3 + (-3)^2 - 17(-3) + 15 = -27 + 9 + 51 + 15 = 48. \]

Since \( P(-3) \neq 0 \), \( x + 3 \) is not a factor.

(2) \( x - 3 \):

If \( x - 3 \) is a factor, then \( c = 3 \). Substitute \( x = 3 \) into \( P(x) \):

\[ P(3) = (3)^3 + (3)^2 - 17(3) + 15 = 27 + 9 - 51 + 15 = 0. \]

Since \( P(3) = 0 \), \( x - 3 \) is a factor.

(3) \( 2x + 3 \):

If \( 2x + 3 \) is a factor, then \( c = -\frac{3}{2} \). Substitute \( x = -\frac{3}{2} \) into \( P(x) \):

\[ P\left(-\frac{3}{2}\right) = \left(-\frac{3}{2}\right)^3 + \left(-\frac{3}{2}\right)^2 - 17\left(-\frac{3}{2}\right) + 15. \]

Simplify step-by-step:

\[ P\left(-\frac{3}{2}\right) = -\frac{27}{8} + \frac{9}{4} + \frac{51}{2} + 15. \]

Convert all terms to have a denominator of 8:

\[ P\left(-\frac{3}{2}\right) = -\frac{27}{8} + \frac{18}{8} + \frac{204}{8} + \frac{120}{8}. \]

\[ P\left(-\frac{3}{2}\right) = \frac{-27 + 18 + 204 + 120}{8} = \frac{315}{8}. \]

Since \( P\left(-\frac{3}{2}\right) \neq 0 \), \( 2x + 3 \) is not a factor.

(4) \( 2x - 3 \):

If \( 2x - 3 \) is a factor, then \( c = \frac{3}{2} \). Substitute \( x = \frac{3}{2} \) into \( P(x) \):

\[ P\left(\frac{3}{2}\right) = \left(\frac{3}{2}\right)^3 + \left(\frac{3}{2}\right)^2 - 17\left(\frac{3}{2}\right) + 15. \]

Simplify step-by-step:

\[ P\left(\frac{3}{2}\right) = \frac{27}{8} + \frac{9}{4} - \frac{51}{2} + 15. \]

Convert all terms to have a denominator of 8:

\[ P\left(\frac{3}{2}\right) = \frac{27}{8} + \frac{18}{8} - \frac{204}{8} + \frac{120}{8}. \]

\[ P\left(\frac{3}{2}\right) = \frac{27 + 18 - 204 + 120}{8} = \frac{-39}{8}. \]

Since \( P\left(\frac{3}{2}\right) \neq 0 \), \( 2x - 3 \) is not a factor.

Final Answer: The correct factor is \( \mathbf{x - 3} \), which corresponds to option \( \mathbf{(2)} \).