Question

Find quadratic polynomial whose sum of zero and products of zero hour respectively 4 and 1.​

Answer 1

$\large {\pmb{\mathfrak{\red{Given...}}}}$

Given that, 4 and 1 are the sum and product of the zeroes of a polynomial respectively.

$\large {\pmb{\mathfrak{\red{To~ find...}}}}$

We have to find the polynomial.

$\large {\pmb{\mathfrak{\red{Solution...}}}}$

According to the question,

• Sum of zeroes = 4 = : $\sf \alpha+\beta$
• Product of zeroes = 1 : $\sf \alpha\beta$

We know that :

$\underline{\pmb{\bf{\green{Polynomial= x^2-(Sum~of~ the ~ zeroes)x+(Product~of ~ the ~zeroes)}}}}$

Now, from the question, we'll substitute the sum and product of the zeroes in the formula above and form the polynomial :

$:\implies \sf x^2-(\alpha+\beta)x+(\alpha\beta)$

$: \implies \sf {x}^{2} - (4)x + (1)$

$: \implies\sf {x}^{2} - 4x + 1$

____________________

$\pmb{\bf{\green{\therefore~x^2-4x+1~is~the~required~polynomial.}}}$

$\large {\pmb{\mathfrak{\red{Important~points...}}}}$

• $\sf \alpha+\beta+\gamma=\dfrac{-b}{a}$

• $\sf \alpha\beta\gamma=\dfrac{-d}{a}$

• $\sf \alpha\beta+\beta\gamma+\gamma\alpha=\dfrac{c}{a}$

• Quadratic formula = $\sf \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$