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4x² - 3x - 1. Find the zeroes of the polynomial, and verify the relation between the coefficients and the zeroes of the polynomial

Solution:

Given, the polynomial is 4x² - 3x - 1.

We have to find the relation between the coefficients and zeros of the polynomial

Let 4x² - 3x - 1 = 0

On factoring,

= 4x² - 4x + x - 1

= 4x(x - 1) + 1(x - 1)

= (4x + 1)(x - 1)

Now, 4x + 1 = 0

4x = -1

x = -1/4

Also, x - 1 = 0

x = 1

Therefore,the zeros of the polynomial are -1/4 and 1.

We know that, if 𝛼 and ꞵ are the zeroes of a polynomial ax² + bx + c, then

Sum of the roots is 𝛼 + ꞵ = -coefficient of x/coefficient of x² = -b/a

Product of the roots is 𝛼ꞵ = constant term/coefficient of x² = c/a

From the given polynomial,

coefficient of x = -3

Coefficient of x² = 4

Constant term = -1

Sum of the roots:

LHS: 𝛼 + ꞵ

= -1/4 + 1

= (-1+4)/4

= 3/4

RHS: -coefficient of x/coefficient of x² = -(-3)/4

= 3/4

LHS = RHS

Product of the roots

LHS: 𝛼ꞵ

= (-1/4)(1)

= -1/4

RHS: constant term/coefficient of x²

= -1/4

LHS = RHS

✦ Try This: Find the zeroes of the polynomial 7x² + 3x - 4, and verify the relation between the coefficients and the zeroes of the polynomial

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 2

NCERT Exemplar Class 10 Maths Exercise 2.3 Problem 1

4x² - 3x - 1. Find the zeroes of the polynomial, and verify the relation between the coefficients and the zeroes of the polynomial

Summary:

The zeroes of the polynomial 4x² - 3x - 1 are -1/4 and 1. The relation between the coefficients and zeros of the polynomial are, Sum of the roots = -b/a = ¾, Product of the roots = c/a = -¼

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