Question

$\alpha$ and $\beta$ are the zeros of a polynomial, such that $\alpha +\beta =6$ and $\alpha \beta =4$. Identify the polynomial.

• A. $x^2-6x+4$
• B. $a^2+6a+4$
• C. $p^2+6p-4$
• D. $m^2-6m-4$

Answer 1

The correct option is A $x^2-6x+4$

Let $p\left(x\right)=a{x}^2+bx+c$ where $a\ne 0$ is a quadratic polynomial

$\alpha ,\beta$ are the zeroes of $p\left(x\right)$

We know that

Sum of the roots $=\alpha +\beta =\frac{-b}{a}$

Product of the roots $=\alpha \times \beta =\frac{c}{a}$

We can also write $p\left(x\right)$ in this form also

$p\left(x\right)=x^2-\frac{-b}{a}x+\frac{c}{a}$

$\Rightarrow p\left(x\right)=x^2-(\alpha +\beta )x+\alpha \beta$

Now given that $\alpha +\beta =6$ and $\alpha \beta =4$

Substituting these values we get

$p\left(x\right)=x^2-\left(6\right)x+4$ is the required polynomial.