Ex.2.4 Q2 Polynomials Solution - NCERT Maths Class 10

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Question

Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as \(2, –7, –14\) respectively.

 Video Solution
Polynomials
Ex 2.4 | Question 2

Text Solution

 

What is known?

Zeroes of a Cubic polynomials are \(2, –7, –14\) respectively.

What is unknown?

A cubic polynomial with the sum, sum of the product of its zeroes taken two at a time.

Reasoning:

To solve this question, follow the steps below-

We know that the general equation of the polynomial is \(a{x^3} + {{ }}b{x^2} + cx{{ }} + {{ }}d\) and the zeroes are \(\alpha ,\;\beta\) and \(\gamma .\)

Now we have to find the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes. For this we know that,

\[\begin{align}{\alpha + \beta + \gamma}& = {\frac{{ - b}}{a}}\\{\alpha \beta + \beta \gamma + \gamma \alpha}& = {\frac{c}{a}}\\{\alpha \beta \gamma}& = \frac{{ - d}}{a}\end{align}\]

Put the values of the known coefficients, you will get the value of unknown coefficient.

Now put the values of coefficients in the general equation of the cubic polynomial \(a{{x}^{3}}+{ }b{{x}^{2}}+cx+d.~\)

Steps:

Let the polynomial be \(a{x^3} + {{ }}b{x^2} + cx + d\) and the zeroes are \(\alpha,\;\beta \)  and \(\gamma.\)

We know that

\[\begin{align}\alpha + \beta + \gamma &= \frac{2}{1}\\ &= \frac{{ - b}}{a}\\\\\alpha \beta + \beta \gamma + \gamma \alpha &= \frac{{ - 7}}{1}\\ &= \frac{c}{a}\\\\\alpha \beta \gamma &= \frac{{ - 14}}{1}\\& = \frac{{ - d}}{a}\end{align}\]

if \(a = 1\) then \(b = - 2, \;c = - 7\) and \(d = 14\)

Hence the polynomial is \({x^3} - 2{x^2} - 7x + 14\)

  
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