Question

(x+a) is a factor of f(x), if

• A. \(f(a)=0\)
• B. \( f(-a)=0\)
• C. \(f(\frac{1}{a})=0\)
• D. \(f(\frac{-1}{a})=0\)

Answer 1

The Correct Option is B

Step 1: Understanding the factor theorem The factor theorem states: If \((x - r)\) is a factor of a polynomial \(f(x)\), then \(f(r) = 0\). This means the value that makes the factor zero is a root of the polynomial.

Step 2: Rewriting the given factor In this question, the factor is \((x + a)\), which can be written as \((x - (-a))\). So, comparing with the factor theorem form, we get: \(r = -a\).

Step 3: Applying the factor theorem Since \(r = -a\), we substitute into the function to check if it's a root: If \((x + a)\) is a factor, then \[ f(-a) = 0 \]

Step 4: Final conclusion Hence, for \((x + a)\) to be a factor of \(f(x)\), it must satisfy \(f(-a) = 0\).

The correct option is (B): \( f(-a)=0\)