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v² + 4√3v - 15. Find the zeroes of the polynomial, and verify the relation between the coefficients and the zeroes of the polynomial

Solution:

Given, the polynomial is v² + 4√3v - 15.

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

Let v² + 4√3v - 15 = 0

On factoring,

v² + 5√3v - √3v - 15 = 0

v(v + 5√3) - √3(v + 5√3) = 0

(v - √3)(v + 5√3) = 0

Now, v - √3 = 0

v = √3

Also, v + 5√3 = 0

v = -5√3

Therefore,the zeros of the polynomial are -5√3 and √3.

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 = 4√3

Coefficient of x² = 1

Constant term = -15

Sum of the roots:

LHS: 𝛼 + ꞵ

= √3 - 5√3

= -4√3

RHS: -coefficient of x/coefficient of x²

= -4√3/1

= -4√3

LHS = RHS

Product of the roots

LHS: 𝛼ꞵ

= (-5√3)(√3)

= -15

RHS: constant term/coefficient of x²

= -15/1

= -15

LHS = RHS

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

v² + 4√3v - 15. 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 v² + 4√3v - 15 are -5√3 and √3. The relation between the coefficients and zeros of the polynomial are, Sum of the roots = -b/a = -4√3, Product of the roots = c/a = -15

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