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y² + (3√5/2)y - 5. Find the zeroes of the polynomial, and verify the relation between the coefficients and the zeroes of the polynomial

Solution:

Given, the polynomial is y² + (3√5/2)y - 5.

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

The polynomial can be rewritten as (1/2)[2y² + 3√5y - 10].

Let 2y² + 3√5y - 10 = 0

On factoring,

2y² + 4√5y - √5y - 10 = 0

2y(y + 2√5) - √5(y - 2√5) = 0

(y + 2√5)(2y - √5) = 0

Now, y + 2√5 = 0

y = -2√5

Also, 2y - √5 = 0

2y = √5

x = √5/2

Therefore,the zeros of the polynomial are √5/2 and -2√5.

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√5

Coefficient of x² = 2

Constant term = -10

Sum of the roots:

LHS: 𝛼 + ꞵ

= √5/2 - 2√5

= (√5 - 4√5)/2

= -3√5/2

RHS: -coefficient of x/coefficient of x²

= -3√5/2

LHS = RHS

Product of the roots

LHS: 𝛼ꞵ

= (√5/2)(-2√5)

= -10/2

= -5

RHS: constant term/coefficient of x²

= -10/2

= -5

LHS = RHS

✦ Try This: Find the zeroes of the polynomial x² + √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 9

y² + (3√5/2)y - 5. 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 y² + (3√5/2)y - 5 are -2√5 and √5/2. The relation between the coefficients and zeros of the polynomial are, Sum of the roots = -b/a = -3√5/2, Product of the roots = c/a = -5

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