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
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) = 2x3 + x2 - 5x + 2
Given zeroes are 1/2, 1, - 2
Substitute x = 1/2 in p (x) = 2x3 + x2 - 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) = 2x3 + x2 - 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) = 2x3 + x2 - 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 x2 / coefficient of x3 [Since the polynomial is 2x3 + x2 - 5x + 2]
αβ + βγ + γα = 1/2 × 1 + 1 × (- 2) + (- 2) × 1/2
= - 5/2
= coefficient of x / coefficient of x3 [Since the polynomial is 2x3 + x2 - 5x + 2]
α.β.γ = 1/2 × 1 × (- 2)
= - 2/2
= - constant term / coefficient of x3 [Since the polynomial is 2x3 + x2 - 5x + 2]
Hence, the relation between zeroes and coefficient is verified.
(ii) x3 - 4x2 + 5x - 2; 2, 1, 1
Given zeroes are 2, 1, 1
Substitute x = 2 in p (x) = x3 - 4x2 + 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 x3 - 4x2 + 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 x2 / coefficient of x3 [Since the polynomial is x3 - 4x2 + 5x - 2]
αβ + βγ + γα = 2 × 1 + 1 × 1 + 1 × 2
= 5
= 5/1
= coefficient of x / coefficient of x3 [Since the polynomial is x3 - 4x2 + 5x - 2]
α.β.γ = 2 × 1 × 1
= 2
= - (-2)/1
= - constant term / coeficient of x3 [Since the polynomial is x3 - 4x2 + 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 2x3 + x2 − 5x + 2 and x3 − 4x2 + 5x − 2 respectively and the relationship between the zeroes and the coefficients have been verified in each case.
☛ Related Questions:
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- If the zeroes of the polynomial x^3 - 3x^2 + x + 1 are a - b, a, a + b, find a and b.
- If two zeroes of the polynomial x^4 - 6x^3 - 26x^2 + 138x - 35 are 2 ± √3 find other zeroes.
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