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.
संबंधित प्रश्न
if α and β are the zeros of ax2 + bx + c, a ≠ 0 then verify the relation between zeros and its cofficients
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
`0, sqrt5`
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
4, 1
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
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
`q(x)=sqrt3x^2+10x+7sqrt3`
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/alpha+1/beta-2alphabeta`
If α and β are the zeroes of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/(aalpha+b)+1/(abeta+b)`.
If 𝛼 and 𝛽 are the zeros of the quadratic polynomial p(x) = 4x2 − 5x −1, find the value of α2β + αβ2.
If α and β are the zeros of the quadratic polynomial f(x) = x2 − px + q, prove that `alpha^2/beta^2+beta^2/alpha^2=p^4/q^2-(4p^2)/q+2`
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and product of its zeros as 3, −1 and −3 respectively.
Verify that 5, -2 and 13 are the zeroes of the cubic polynomial `p(x) = (3x^3 – 10x^2 – 27x + 10)` and verify the relation between its zeroes and coefficients.
If 3 and –3 are two zeroes of the polynomial `(x^4 + x^3 – 11x^2 – 9x + 18)`, find all the zeroes of the given polynomial.
If x + 2 is a factor of x2 + ax + 2b and a + b = 4, then
Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is ______.
If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.
For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.
`(-8)/3, 4/3`
Given that `sqrt(2)` is a zero of the cubic polynomial `6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2)`, find its other two zeroes.
Find the sum and product of the roots of the quadratic equation 2x2 – 9x + 4 = 0.
The zeroes of the polynomial p(x) = 2x2 – x – 3 are ______.
