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4x² + 5√2x - 3. 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 + 5√2x - 3.

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

Let 4x² + 5√2x - 3 = 0

On factoring,

= 4x² + 6√2x - √2x - 3

= 2√2x(√2x + 3) - (√2x + 3)

= (2√2x - 1)(√2x + 3)

Now, 2√2x - 1 = 0

2√2x = 1

x = 1/2√2

Also, √2x + 3 = 0

√2x = -3

x = -3/√2

Therefore,the zeros of the polynomial are 1/2√2 and -3/√2.

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 = 5√2

Coefficient of x² = 4

Constant term = -3

Sum of the roots:

LHS: 𝛼 + ꞵ

= 1/2√2 - 3/√2

= (1-6)/2√2

= -5/2√2

= -5√2/4

RHS: -coefficient of x/coefficient of x²

= -5√2/4

LHS = RHS

Product of the roots

LHS: 𝛼ꞵ

= (-3/√2)(1/2√2)

= -3/4

RHS: constant term/coefficient of x²

= -3/4

LHS = RHS

Therefore, the zeroes of the polynomial are -3/√2 and 1/2√2. The relation between the coefficients and zeros of the polynomial are, Sum of the roots = -b/a = -5√2/4, Product of the roots = c/a = -¾.

✦ Try This: Find the zeroes of the polynomial 4x² + 3√2x - 8, 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 6

4x² + 5√2x - 3. 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 + 5√2x - 3 are -3/√2 and 1/2√2. The relation between the coefficients and zeros of the polynomial are, Sum of the roots = -b/a = -5√2/4, Product of the roots = c/a = -¾

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