# Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients in each case:

(i) 2x^{3} + x^{2} - 5x + 2; 1/2, 1, - 2

(ii) x^{3} - 4x^{2} + 5x - 2; 2, 1, 1

**Solution:**

For the given cubic polynomials, we will substitute the values of zeroes in the polynomial to check if it satisfies the polynomial followed by verifying the relationship between the zeroes and the coefficients.

(i) p (x) = 2x^{3} + x^{2} - 5x + 2

Given zeroes are 1/2, 1, - 2

Substitute x = 1/2 in p (x) = 2x^{3} + x^{2} - 5x + 2

p (1/2) = 2 (1/2)^{3} + (1/2)^{2} - 5 (1/2) + 2

p (1/2) = 2 (1/8) + (1/4) - 5/2 + 2

p (1/2) = 1/4 + 1/4 - 5/2 + 2

p (1/2) = (1 + 1 - 10 + 8)/4

p (1/2) = 0

Substitute x = 1 in p (x) = 2x^{3} + x^{2} - 5x + 2

p (1) = 2 (1)^{3} + (1)^{2 }- 5 (1) + 2

p (1) = 2 + 1 - 5 + 2

p (1) = 0

Substitute x = - 2 in p (x) = 2x^{3} + x^{2} - 5x + 2

p (- 2) = 2 (- 2)^{3} + (- 2)^{2} - 5 (- 2) + 2

p (- 2) = -16 + 4 + 10 + 2

p (- 2) = -16 + 16

p (- 2) = 0

Therefore, 1/2, 1, - 2 are the zeroes of the polynomial.

Now let α = 1/2, β = 1 and γ = 2

α + β + γ = 1/2 + 1 + (- 2)

= - 1/2

= - coefficient of x^{2 }/ coefficient of x^{3} [Since the polynomial is 2x^{3} + x^{2} - 5x + 2]

αβ + βγ + γα = 1/2 × 1 + 1 × (- 2) + (- 2) × 1/2

= - 5/2

= coefficient of x / coefficient of x^{3} [Since the polynomial is 2x^{3} + x^{2} - 5x + 2]

α.β.γ = 1/2 × 1 × (- 2)

= - 2/2

= - constant term / coefficient of x^{3} [Since the polynomial is 2x^{3} + x^{2} - 5x + 2]

Hence, the relation between zeroes and coefficient is verified.

(ii) x^{3} - 4x^{2} + 5x - 2; 2, 1, 1

Given zeroes are 2, 1, 1

Substitute x = 2 in p (x) = x^{3} - 4x^{2} + 5x - 2

p (2) = (2)^{3} - 4(2)^{2} + 5(2) - 2

p (2) = 8 - 16 + 10 - 2

p (2) = 18 - 18

p (2) = 0

Substitute x = 1 in x^{3} - 4x^{2} + 5x - 2

p (1) = (1)^{3} - 4(1)^{2} + 5(1) - 2

p (1) = 1 - 4 + 5 - 2

p (1) = - 3 + 3

p (1) = 0

Therefore, 2,1 and 1 are the zeroes of the polynomial.

Now let α = 2, β = 1 and γ = 1

α + β + γ = 2 + 1 + 1

= 4/1

= - coefficient of x^{2} / coefficient of x^{3} [Since the polynomial is x^{3} - 4x^{2} + 5x - 2]

αβ + βγ + γα = 2 × 1 + 1 × 1 + 1 × 2

= 5

= 5/1

= coefficient of x / coefficient of x^{3} [Since the polynomial is x^{3} - 4x^{2} + 5x - 2]

α.β.γ = 2 × 1 × 1

= 2

= - (-2)/1

= - constant term / coeficient of x^{3} [Since the polynomial is x^{3} - 4x^{2} + 5x - 2]

Hence, the relation between zeroes and coefficient is verified.

**☛ Check: **NCERT Solutions Class 10 Maths Chapter 2

**Video Solution:**

## Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients in each case: (i) 2x³ + x² - 5x + 2 ; 1/2, 1, - 2 (ii) x³ - 4x² + 5x - 2 ; 2, 1, 1

NCERT Solutions Class 10 Maths Chapter 2 Exercise 2.4 Question 1:

**Summary:**

For the numbers given alongside of the cubic polynomials along with their zeroes we see that 1/2, 1, -2, and 2,1,1 are the zeroes of the polynomials 2x^{3 }+ x^{2 }− 5x + 2 and x^{3 }− 4x^{2 }+ 5x − 2 respectively and the relationship between the zeroes and the coefficients have been verified in each case.

**☛ Related Questions:**

- Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, - 7, - 14 respectively.
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- If two zeroes of the polynomial x^4 - 6x^3 - 26x^2 + 138x - 35 are 2 ± √3 find other zeroes.
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