# 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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