Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Given, `sqrt(2)` is one of the zero of the cubic polynomial.
Then, `(x - sqrt(2))` is one of the factor of the given polynomial p(x) = `6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2)`.
So, by dividing p(x) by `x - sqrt(2)`
`6x^2 + 7sqrt(2)x + 4`
`(x - sqrt(2))")"overline(6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2))`
`6x^3 - 6sqrt(2)x^2`
– +
`7sqrt(2)x^2 - 10x - 4sqrt(2)`
`7sqrt(2)x^2 - 14x`
– +
`4x - 4sqrt(2)`
`4x - 4sqrt(2)`
0
`6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2) = (x - sqrt(2)) (6x^2 + 7sqrt(2)x + 4)`
By splitting the middle term,
We get,
`(x - sqrt(2)) (6x^2 + 4sqrt(2)x + 3sqrt(2)x + 4)`
= `(x - sqrt(2)) [2x(3x + 2sqrt(2)) + sqrt(2)(3x + 2sqrt(2))]`
= `(x - sqrt(2)) (2x + sqrt(2)) (3x + 2sqrt(2))`
To get the zeroes of p(x),
Substitute p(x) = 0
`(x - sqrt(2)) (2x + sqrt(2)) (3x + 2sqrt(2))` = 0
`x = sqrt(2) , x = -sqrt(2)/2, x = (-2sqrt(2))/3`
Hence, the other two zeroes of p(x) are `-sqrt(2)/2` and `(-2sqrt(2))/3`.
APPEARS IN
संबंधित 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`
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.
6x2 – 3 – 7x
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
4, 1
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/alpha-1/beta`
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 f(x) = 6x2 + x − 2, find the value of `alpha/beta+beta/alpha`.
If 𝛼 and 𝛽 are the zeros of the quadratic polynomial p(x) = 4x2 − 5x −1, find the value of α2β + αβ2.
Find the zeroes of the polynomial f(x) = `2sqrt3x^2-5x+sqrt3` and verify the relation between its zeroes and coefficients.
Find the zeroes of the quadratic polynomial` (x^2 ˗ 5)` and verify the relation between the zeroes and the coefficients.
If 2 and -2 are two zeroes of the polynomial `(x^4 + x^3 – 34x^2 – 4x + 120)`, find all the zeroes of the given polynomial.
If 𝛼, 𝛽 are the zeroes of the polynomial `f(x) = 5x^2 -7x + 1` then `1/∝+1/β=?`
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 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.
`21/8, 5/16`
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.
`-2sqrt(3), -9`
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`v^2 + 4sqrt(3)v - 15`
Find the zeroes of the quadratic polynomial 6x2 – 3 – 7x and verify the relationship between the zeroes and the coefficients.
If α and β are the zeros of a polynomial f(x) = px2 – 2x + 3p and α + β = αβ, then p is ______.
If the zeroes of the polynomial x2 + px + q are double in value to the zeroes of the polynomial 2x2 – 5x – 3, then find the values of p and q.
If α, β are zeroes of quadratic polynomial 5x2 + 5x + 1, find the value of α–1 + β–1.
