Question

If \( (x+3) \) is a factor of \( ax^2 + x + 1 \), then the value of \( a \) is:

• A. \( 3 \)
• B. \( \frac{9}{2} \)
• C. \( \frac{2}{9} \)
• D. \( 9 \)

Hint:

The factor theorem states that if \( (x + k) \) is a factor of a polynomial \( f(x) \), then \( f(-k) = 0 \).

Answer 1

The Correct Option is C

Step 1: Use the factor theorem If \( (x+3) \) is a factor, then substituting \( x = -3 \) in \( ax^2 + x + 1 \) should yield 0. Step 2: Substitute \( x = -3 \) \[ a(-3)^2 + (-3) + 1 = 0 \] \[ 9a - 3 + 1 = 0 \] \[ 9a - 2 = 0 \] \[ 9a = 2 \] \[ a = \frac{2}{9} \] Step 3: Conclusion Thus, the correct answer is \( \frac{2}{9} \).