VERIFY THE RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS

Find the zeros of the following polynomials by factorization method and verify the relations between the zeros and the coefficients of the polynomials:

Problem 1 :

4x² - 3x - 1 = 0

Solution :

4x² - 3x - 1 = 0

4x² - 4x + x - 1 = 0

4x(x - 1) + 1(x - 1) = 0

(4x + 1) (x - 1) = 0

4x + 1 = 0   x - 1 = 0

x = -1/4     x = 1

So, α = -1/4 and β = 1

Verifying relationship :

4x² - 3x - 1 = 0

ax² + bx + c = 0

a = 4, b = -3, c = -1

Verified.

Problem 2 :

3x² + 4x - 4 = 0

Solution :

First, let us find the zeros of the polynomial.

3x² + 4x - 4 = 0

3x² + 6x - 2x - 4 = 0

3x(x + 2) - 2(x + 2) = 0

(3x - 2) (x + 2) = 0

3x - 2 = 0   x + 2 = 0

x = 2/3      x = -2

So, α = 2/3 and β = -2

Verifying relationship :

3x² + 4x - 4 = 0

ax² + bx + c = 0

a = 3, b = 4, c = -4

Hence, verified the relations between the zeros and the coefficients of the polynomial.

Problem 3 :

5t² + 12t + 7 = 0

Solution :

First, let us find the zeros of the polynomial.

5t² + 12t + 7 = 0

5t² + 5t + 7t + 7 = 0

5t(t + 1) + 7(t + 1) = 0

(5t + 7) (t + 1) = 0

5t + 7 = 0   t + 1 = 0

t = -7/5      t = -1

So, α = -7/5 and β = -1

Verifying relationship :

5t² + 12t + 7 = 0

ax² + bx + c = 0

a = 5, b = 12, c = 7

Verified.