Question

If \[ f(x)=x^2+5x+p \] and \[ g(x)=x^2+3x+q \] have a common factor, then \[ (p-q)^2= \]

• A. \(4(2p+3q)\)
• B. \(3(2p-3q)\)
• C. \(2(3p-2q)\)
• D. \(2(3p-5q)\)

Hint:

For common-factor problems, assume a common root and use substitution.

Answer 1

The Correct Option is D

Concept: If two quadratic polynomials have a common factor, they share a common root. Let the common root be: \[ \alpha \] Then: \[ \alpha^2+5\alpha+p=0 \] \[ \alpha^2+3\alpha+q=0 \]

Step 1: Subtract the equations.
\[ (\alpha^2+5\alpha+p)-(\alpha^2+3\alpha+q)=0 \] \[ 2\alpha+p-q=0 \] \[ \alpha=\frac{q-p}{2} \]

Step 2: Substitute into one equation.
Using: \[ \alpha^2+3\alpha+q=0 \] Substitute: \[ \left(\frac{q-p}{2}\right)^2+3\left(\frac{q-p}{2}\right)+q=0 \] Multiply by \(4\): \[ (q-p)^2+6(q-p)+4q=0 \] Rearrange: \[ (p-q)^2=6p-10q \] Factor out \(2\): \[ (p-q)^2=2(3p-5q) \] Thus, the required answer is: \[ \boxed{2(3p-5q)} \]