Ex.2.2 Q2 Polynomials Solution - NCERT Maths Class 10
Question
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
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(i) \(\begin{align}\frac{1}{4},\, - 1\end{align}\) |
(ii) \(\begin{align}\sqrt 2 ,\;\frac{1}{3}\end{align}\) |
(iii) \({0,\;5}\) |
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(iv) \(1, \;1\) |
(v) \(\begin{align}\frac{1}{4},\;\frac{1}{4} \end{align}\) |
(vi) \(4,\;1\) |
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:
\[\begin{align}&k\left( {{x^2}-\left( {{\text{sum of roots}}} \right)x + {\text{ product of roots}}} \right)\\&k\left\{ {{x^2}-{\text{ }}4x + {\text{ }}1} \right\}\end{align}\]