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t³ - 2t² - 15t. Find the zeroes of the polynomial, and verify the relation between the coefficients and the zeroes of the polynomial

Solution:

Given, the polynomial is t³ - 2t² - 15t.

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

Let t³ - 2t² - 15t = 0

Taking out common term,

t(t² - 2t - 15) = 0

t = 0

On factoring,

t² - 2t - 15 = t² - 5t + 3t - 15

= t(t - 5) + 3(t - 5)

= (t + 3)(t - 5)

Now, t + 3 = 0

t = -3

Also, t - 5 = 0

t = 5

Therefore,the zeros of the polynomial are 0, -3 and 5.

We know that, if 𝛼, ꞵ and 𝛾 are the zeroes of a cubic polynomial ax³ + bx² + cx + d, then

𝛼 + ꞵ + 𝛾 = -b/a = -coefficient of x²/coefficient of x³

𝛼ꞵ + ꞵ𝛾 + 𝛾𝛼 = c/a = coefficient of x/coefficient of x³

𝛼ꞵ𝛾 = -d/a = -constant/coefficient of x³

From the given polynomial,

Coefficient of x³ = 1

coefficient of x² = -2

coefficient of x = -15

Constant term = 0

Sum of the roots:

LHS: 𝛼 + ꞵ + 𝛾

= 0 - 3 + 5

= 5 - 3

= 2

RHS: -coefficient of x²/coefficient of x³

= -(-2)/1

= 2

LHS = RHS

Sum of the products of two roots at a time:

LHS: 𝛼ꞵ + ꞵ𝛾 + 𝛾𝛼

= (0)(-3) + (-3)(5) + (5)(0)

= 0 - 15 + 0

= -15

RHS: coefficient of x/coefficient of x³

= -15/1

= -15

LHS = RHS

Product of all the roots

LHS: 𝛼ꞵ𝛾

= (0)(-3)(5)

= 0

RHS: -constant/coefficient of x³

= -0/1

= 0

LHS = RHS

✦ Try This: Find the zeroes of the polynomial 3t³ - 2t² - 5t, 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 4

t³ - 2t² - 15t. 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 t³ - 2t² - 15t are 0, -3 and 5. The relation between the coefficients and zeros of the polynomial are Sum of the roots = -b/a = 2 Sum of the product of two roots at a time = c/a = -15 Product of all the roots = -d/a = 0

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