Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time 3, -1 3

Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and product of its zeros as 3, −1 and −3 respectively.

Any cubic polynomial is of the form ax3 + bx2 + cx + d = x3 − sum of zeroes (x2)[product of zeroes] + sum of the products of its zeroes × - product of zeroes

= 𝑥3 − 2𝑥2 + (3 − 𝑥) + 3

= k [𝑥3 − 3𝑥2 − 𝑥 − 3]

k is any non-zero real numbers

Concept: Relationship Between Zeroes and Coefficients of a Polynomial

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Solution:

We know that the general form of a cubic polynomial is ax3 + bx2 + cx + d and the zeroes are α, β, and γ.

Let's look at the relation between sum, and product of its zeroes and coefficients of the polynomial.

• α + β + γ = - b / a
• αβ + βγ + γα = c / a
• α x β x γ = - d / a

Let the polynomial be ax3 + bx2 + cx + d and the zeroes are α, β, γ

We know that,

α + β + γ = 2/1 = - b / a

αβ + βγ + γα = - 7/1 = c / a

α.β.γ = - 14/1 = - d / a

Thus, by comparing the coefficients we get, a = 1, then b = - 2, c = - 7 and d = 14

Now, substitute the values of a, b, c, and d in the cubic polynomial ax3 + bx2 + cx + d.

Hence the polynomial is x3 - 2x2 - 7x + 14.

☛ Check: NCERT Solutions Class 10 Maths Chapter 2

Video Solution:

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

NCERT Solutions Class 10 Maths Chapter 2 Exercise 2.4 Question 2

Summary:

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 is x3 - 2x2 - 7x + 14.

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Find the cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and product of its zeros as 3, 1 and 3 respectively.