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.
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
Find the zeros of the quadratic polynomial 6x2 - 13x + 6 and verify the relation between the zero and its coefficients.
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
`g(x)=a(x^2+1)-x(a^2+1)`
Find the zeroes of the quadratic polynomial `(3x^2 ˗ x ˗ 4)` and verify the relation between the zeroes and the coefficients.
If 𝛼, 𝛽 are the zeroes of the polynomial `f(x) = 5x^2 -7x + 1` then `1/∝+1/β=?`
If α, β are the zeros of the polynomial f(x) = ax2 + bx + c, then\[\frac{1}{\alpha^2} + \frac{1}{\beta^2} =\]
If one of the zeroes of the quadratic polynomial (k – 1)x2 + k x + 1 is –3, then the value of k is ______.
The number of polynomials having zeroes as –2 and 5 is ______.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
5t2 + 12t + 7
Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.
