Question

If \((x-2)\) is a common factor of the expressions \(x^2 + ax + b\) and \(x^2 + cx + d\), then \(\dfrac{b-d}{\,c-a\,}\) is equal to:

• A. \(-2\)
• B. \(-1\)
• C. \(1\)
• D. \(2\)

Hint:

If ((x-k)) is a factor of a polynomial (f(x)), then always use the condition (f(k)=0) to form equations.

Answer 1

The Correct Option is D

Step 1: Use the factor condition. Since \((x-2)\) is a common factor of both expressions, substituting \(x=2\) makes each expression zero. Step 2: Apply the condition to the first expression. \[ 2^2 + 2a + b = 0 \] \[ 4 + 2a + b = 0 \] \[ b = -4 - 2a \qquad \cdots (1) \] Step 3: Apply the condition to the second expression. \[ 2^2 + 2c + d = 0 \] \[ 4 + 2c + d = 0 \] \[ d = -4 - 2c \qquad \cdots (2) \] Step 4: Find \(b-d\). Using (1) and (2): \[ b - d = (-4 - 2a) - (-4 - 2c) \] \[ b - d = -2a + 2c = 2(c-a) \] Step 5: Evaluate the required expression. \[ \dfrac{b-d}{c-a} = \dfrac{2(c-a)}{c-a} = 2 \] Step 6: Final conclusion. \[ \boxed{2} \]