# Ex.2.2 Q2 Polynomials Solution - NCERT Maths Class 10

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## Question

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

 (i) \begin{align}\frac{1}{4},\, - 1\end{align} (ii) \begin{align}\sqrt 2 ,\;\frac{1}{3}\end{align} (iii) $${0,\;5}$$ (iv) $$1, \;1$$ (v)  \begin{align}\frac{1}{4},\;\frac{1}{4} \end{align} (vi)  $$4,\;1$$

Video Solution
Polynomials
Ex 2.2 | Question 2

## Text Solution

What is known?

The sum and product of zeroes of quadratic polynomials.

What is unknown?

A quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.

Reasoning:

This question is straight forward - the value of sum of roots and product of roots is given. You have to form a quadratic polynomial. Put the values in the general equation of the quadratic polynomial i.e.

$k\left( {{x^2}-{\text{ }}\left( {{\text{sum of roots}}} \right)x{\text{ }} + {\text{ product of roots}}} \right)$

Steps:

(i) \begin{align}\,\,\frac{1}{4},\; - 1\end{align}

We know that the general equation of a quadratic polynomial is:

\begin{align}&{k\left( {{x^2} - \left( {{\text{sum of roots}}} \right){\text{ }}x + {\text{ product of roots }}} \right)}\\&{k\left\{ {{x^2} - \frac{1}{4}x + \frac{1}{4} \times - 1} \right\}}\\&{k\left\{ {{x^2} - \frac{1}{4}x - \frac{1}{4}} \right\}}\end{align}

(ii)  \begin{align}\sqrt 2,\;\frac{1}{3}\end{align}

We know that the general equation of a quadratic polynomial is:

\begin{align}&k\left( {{x^2} - \left( {{\text{sum}}\,\,{\text{of}}\,\,{\text{roots}}} \right)x + {\text{product}}\,{\text{of}}\,{\text{roots}}} \right)\\&k\left\{ {{x^2} - \sqrt 2 x + \frac{1}{3}} \right\}\end{align}

(iii)  $$\,0,\;\sqrt 5$$

We know that the general equation of a quadratic polynomial is

\begin{align}&k\left( {{x^2}-{\text{ }}\left( {{\text{sum of roots}}} \right)x{\text{ }} + {\text{ product of roots}}} \right)\\&k\left\{ {{x^2}-{\text{ }}0.x{\text{ }} + {\text{ }}\sqrt 5 } \right\}\\ &k\left\{ {{x^2}+ {\text{ }}\sqrt 5 } \right\}\end{align}

(iv) $$\,1,\;1$$

We know that the general equation of a quadratic polynomial is

\begin{align}&k\left( {{x^2}-{\text{ }}\left( {{\text{sum of roots}}} \right)x{\text{ }} + {\text{ product of roots}}} \right)\\&k\left\{ {{x^2}-{\text{ }}1x{\text{ }} + {\text{ }}1} \right\}\\&k\left\{ {{x^2}-{\text{ }}x{\text{ }} + {\text{ }}1} \right\}\end{align}

(v) \begin{align} - \frac{1}{4},\;\frac{1}{4}\end{align}

We know that the general equation of a quadratic polynomial is:

\begin{align}& k\left( {{x^2} - \left( {{\text{sum}}\,\,{\text{of}}\,{\text{roots}}} \right)\,x + {\text{product}}\,{\text{of}}\,{\text{roots}}} \right)\\&k\left( {{x^2} - \left( { - \frac{1}{4}x} \right) + \frac{1}{4}} \right) \\& k\left( {{x^2} + \frac{1}{4}x + \frac{1}{4}} \right) \\ \end{align}

(vi)  $$4,\;1$$

We know that the general equation of a quadratic polynomial is:

\begin{align}&k\left( {{x^2}-\left( {{\text{sum of roots}}} \right)x + {\text{ product of roots}}} \right)\\&k\left\{ {{x^2}-{\text{ }}4x + {\text{ }}1} \right\}\end{align}

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