Question

If $Q_1(x) = x^2 + (k-29)x - k$ and $Q_2(x) = 2x^2 + (2k-43)x + k$ both are factors of cubic polynomial, then largest value of $k$ is (where $Q_1(x)$ and $Q_2(x)$ are not perfect squares)

• A. 0
• B. 33
• C. 23
• D. 30

Answer 1

Answer: (D)

30

Answer 2

For two quadratic polynomials to be factors of a cubic polynomial, they must share a common linear factor, which means they share a common root. By setting up equations $Q_1(\alpha)=0$ and $Q_2(\alpha)=0$ for a common root $\alpha$, we solved for $\alpha$ in terms of $k$, yielding $\alpha = -k/5$. Substituting this back into $Q_1(\alpha)=0$ resulted in a quadratic equation in $k$, $-4k^2 + 120k = 0$, which gives $k=0$ or $k=30$. Finally, we verified that for both these values of $k$, the discriminants of $Q_1(x)$ and $Q_2(x)$ are non-zero, satisfying the condition that they are not perfect squares. The largest of the valid $k$ values is 30.