Advertisements
Advertisements
प्रश्न
Verify that the numbers given along side of the cubic polynomials are their zeroes. Also verify the relationship between the zeroes and the coefficients.
`2x^3 + x^2 – 5x + 2 ; 1/2, 1, – 2`
Advertisements
उत्तर
p(x) = 2x3 + x2 - 5x + 2
Zeroes for this polynomial are 1/2, 1, -2
p(1/2) = `2(1/2)^3+(1/2)^2-5(1/2)+2`
= `1/4+1/4-5/2+2`
=0
p(1) = 2 x 13 + 12 - 5 x 1 + 2
= 0
p(-2) = 2(-2)3 + (-2)2 - 5(-2) + 2
= -16 + 4 + 10 + 2 = 0
Therefore, 1/2, 1, and −2 are the zeroes of the given polynomial.
Comparing the given polynomial with ax3 + bx2 + cx + d we obtain a = 2, b = 1, c = −5, d = 2
We can take α = 1/2, β = 1, y = -2
α + β + γ = `1/2+1+(-2) = -1/2 = (-b)/a`
αβ + βγ + αγ = `1/2xx1+1(-2)+1/2(-2)=(-5)/2 = c/a`
αβγ = `1/2xx1xx(-2) = (-1)/1=(-(2))/2=(-d)/a`
Therefore, the relationship between the zeroes and the coefficients is verified.
APPEARS IN
संबंधित प्रश्न
Find the zeros of the quadratic polynomial 4x2 - 9 and verify the relation between the zeros and its coffiecents.
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.
4u2 + 8u
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
1, 1
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α4 + β4
If the zeros of the polynomial f(x) = 2x3 − 15x2 + 37x − 30 are in A.P., find them.
Find the zeroes of the quadratic polynomial `(8x^2 ˗ 4)` and verify the relation between the zeroes and the coefficients
Define a polynomial with real coefficients.
If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, the\[\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} =\]
If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, then α2 + β2 + γ2 =
If α, β are the zeros of the polynomial f(x) = ax2 + bx + c, then\[\frac{1}{\alpha^2} + \frac{1}{\beta^2} =\]
If two zeros x3 + x2 − 5x − 5 are \[\sqrt{5}\ \text{and} - \sqrt{5}\], then its third zero is
If two zeroes of the polynomial x3 + x2 − 9x − 9 are 3 and −3, then its third zero is
The below picture are few natural examples of parabolic shape which is represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.
If the sum of the roots is –p and the product of the roots is `-1/"p"`, then the quadratic polynomial is:
If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it ______.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`7y^2 - 11/3 y - 2/3`
Find the zeroes of the quadratic polynomial 6x2 – 3 – 7x and verify the relationship between the zeroes and the coefficients.
If one zero of the polynomial p(x) = 6x2 + 37x – (k – 2) is reciprocal of the other, then find the value of k.
If α, β are zeroes of the quadratic polynomial x2 – 5x + 6, form another quadratic polynomial whose zeroes are `1/α, 1/β`.
Find a quadratic polynomial whose zeroes are 6 and – 3.
The zeroes of the polynomial p(x) = 2x2 – x – 3 are ______.
