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2x² + (7/2)x + (3/4). Find the zeroes of the polynomial, and verify the relation between the coefficients and the zeroes of the polynomial

Solution:

Given, the polynomial is 2x² + (7/2)x + (3/4)

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

The polynomial can be rewritten as (1/4)[8x² + 14x + 3]

Let (1/4)[8x² + 14x + 3] = 0

8x² + 14x + 3 = 0

On factoring,

8x² + 14x + 3 = 8x² + 12x + 2x + 3

= 4x(2x + 3) + (2x + 3)

= (4x + 1)(2x + 3)

Now, (2x + 3) = 0

2x = -3

x = -3/2

4x + 1 = 0

4x = -1

x = -1/4

Therefore,the zeros of the polynomial are -3/2 and -1/4.

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 = 14

Coefficient of x² = 8

Constant term = 3

Sum of the roots:

LHS: 𝛼 + ꞵ

= -1/4 + (-3/2)

= (-2-12)/8

= -14/8

= -7/4

RHS: -coefficient of x/coefficient of x²

= -14/8

= -7/4

LHS = RHS

Product of the roots

LHS: 𝛼ꞵ

= (-1/4)(-3/2)

= 3/8

RHS: constant term/coefficient of x²

= 3/8

LHS = RHS

✦ Try This: Find the zeroes of the polynomial 2x² + (7/4)x + 3, 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 5

2x² + (7/2)x + (3/4). 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 2x² + (7/2)x + (3/4) are -1/4 and -3/2 and the relation between the coefficients and zeros of the polynomial are, Sum of the roots = -b/a = -7/4, Product of the roots = c/a = ⅜

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