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# Which of the following quadratic polynomial having zeros $1$ and $-2$: A) ${{x}^{2}}-x+2$ B) ${{x}^{2}}-x-2$C) ${{x}^{2}}+x-2$D) None of these

Last updated date: 23rd Jul 2024
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Hint: The given question is related to quadratic equations. Try to recall the formulae related to the relation between the coefficients and sum and product of the roots of a quadratic equation.

Before proceeding with the solution, we must know about the relation between the coefficients and sum and product of the roots of the quadratic equation given by $a{{x}^{2}}+bx+c=0$ .
We know, the roots of the equation $a{{x}^{2}}+bx+c=0$ are given by the quadratic formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .
Let $\alpha$ and $\beta$ be the roots of the equation. So, $\alpha =\dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}$ and $\beta =\dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$. The sum of the roots is given as $\alpha +\beta =\left( \dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a} \right)+\left( \dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a} \right)=\dfrac{-2b}{2a}=\dfrac{-b}{a}$ .
So, the sum of the roots is related to the coefficients as $\alpha +\beta =\dfrac{-b}{a}$ .
The product of the roots is given as $\alpha \beta =\left( \dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a} \right)\left( \dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a} \right)=\dfrac{{{b}^{2}}-\left( {{b}^{2}}-4ac \right)}{4{{a}^{2}}}=\dfrac{c}{a}$ .
So, the product of the roots is related to the coefficients as $\alpha \beta =\dfrac{c}{a}$ .
Now, we have $a{{x}^{2}}+bx+c=0$. On dividing the equation by $a$ , we get ${{x}^{2}}+\dfrac{b}{a}x+\dfrac{c}{a}=0.....(i)$.
We have $\alpha +\beta =\dfrac{-b}{a}$ and $\alpha \beta =\dfrac{c}{a}$ . So, we can rewrite equation $(i)$ with coefficients in the form sum and product of roots as ${{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta =0$.
Now, coming to the question , we are given the zeros of a quadratic polynomial as $1$ and $-2$. So, the sum of zeroes is equal to $-2+1=-1$ and the product of zeroes is equal to $-2\times 1=-2$ .
Hence, the quadratic polynomial having zeros $1$ and $-2$ is given as ${{x}^{2}}+x-2$ .
Note: The quadratic equation with coefficients in the form sum and product of roots is given as ${{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta =0$ and not ${{x}^{2}}+\left( \alpha +\beta \right)x+\alpha \beta =0$. Students often get confused and make a mistake. Such mistakes should be avoided as they can lead to wrong answers.