Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let `x^4 + x^3 – 11x^2 – 9x + 18`
Since 3 and – 3 are the zeroes of f(x), it follows that each one of (x + 3) and (x – 3) is a factor of f(x).
Consequently, `(x – 3) (x + 3) = (x^2 – 9)` is a factor of f(x).
On dividing `f(x) by (x^2 – 9)`, we get:
`f(x) = 0 ⇒ (x^2 + x – 2) (x^2 – 9) = 0`
⇒ `(x^2 + 2x – x – 2) (x – 3) (x + 3)`
⇒ `(x – 1) (x + 2) (x – 3) (x + 3) = 0`
⇒ `x = 1 or x = -2 or x = 3 or x = -3`
Hence, all the zeroes are 1, -2, 3 and -3.
APPEARS IN
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 the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.
4u2 + 8u
If two zeroes of the polynomial x4 – 6x3 – 26x2 + 138x – 35 are 2 ± `sqrt3` , find other zeroes
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 the sum of the zeros of the quadratic polynomial f(t) = kt2 + 2t + 3k is equal to their product, find the value of k.
If α and β are the zeroes of the polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α − β)2.
Find the zeroes of the quadratic polynomial `2x^2 ˗ 11x + 15` and verify the relation between the zeroes and the coefficients.
Find the zeroes of the quadratic polynomial `(5y^2 + 10y)` and verify the relation between the zeroes and the coefficients.
Find the quadratic polynomial whose zeroes are `2/3` and `-1/4` Verify the relation between the coefficients and the zeroes of the polynomial.
Find a cubic polynomial whose zeroes are `1/2, 1 and -3.`
Find a cubic polynomial with the sum of its zeroes, sum of the products of its zeroes taken two at a time and the product of its zeroes as 5, -2 and -24 respectively.
Define a polynomial with real coefficients.
The product of the zeros of x3 + 4x2 + x − 6 is
If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign.
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.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`2s^2 - (1 + 2sqrt(2))s + sqrt(2)`
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`
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`
The only value of k for which the quadratic polynomial kx2 + x + k has equal zeros is `1/2`
