Ex.2.2 Q2 Polynomials Solution - NCERT Maths Class 10

Go back to  'Ex.2.2'

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:

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

  
Learn math from the experts and clarify doubts instantly

  • Instant doubt clearing (live one on one)
  • Learn from India’s best math teachers
  • Completely personalized curriculum