Advertisements
Advertisements
Question
If a and 3 are the zeros of the quadratic polynomial f(x) = x2 + x − 2, find the value of `1/alpha-1/beta`.
Advertisements
Solution 1
Since 𝛼 and 𝛽 are the roots of the polynomial x + x – 2
∴ Sum of roots α + β = 1
Product of roots αβ 2 ⇒ `-1/beta`
`=(beta-alpha)/alphabeta*(alpha-beta)/alphabeta`
`=(sqrt((alpha+beta)^2-4alphabeta))/(alphabeta)`
`=sqrt(1+8)/(+2)`
`=3/2`
Solution 2
Given if α and β are the solutions of the polynomial f(x) = x2 + x − 2.
So, first let us find zeros of f(x) = 0:
The middle term x is expressed as sum of 2x and −x such that its product is equals to product of extreme terms.
(-2) x x2 = -2x2
Thus, x2 + 2x - x - 2 = 0
x(x + 2) - 1(x + 2) = 0
(x + 2)(x - 1) = 0
(x + 2) = 0 or (x - 1) = 0
=> x = -2 or x = 1
∴ α, β = (1, -2) or (-2, 1)
Case 1: When (α, β) = (1, -2)
`(1/alpha - 1/beta) = 1/1 - 1/(-2)`
= `1 + 1/2`
= `(2 + 1)/2`
∴ `1/alpha - 1/beta = (-3)/2`
Hence, `1/alpha - 1/beta = (-3)/2 or 3/2`
APPEARS IN
RELATED QUESTIONS
Find the zeros of the quadratic polynomial 6x2 - 13x + 6 and verify the relation between the zero and its coefficients.
if α and β are the zeros of ax2 + bx + c, a ≠ 0 then verify the relation between zeros and its cofficients
If a and are the zeros of the quadratic polynomial f(x) = 𝑥2 − 𝑥 − 4, find the value of `1/alpha+1/beta-alphabeta`
If α and β are the zeros of the quadratic polynomial f(x) = x2 − 1, find a quadratic polynomial whose zeroes are `(2alpha)/beta" and "(2beta)/alpha`
If α and β are the zeros of the quadratic polynomial f(x) = x2 − p (x + 1) — c, show that (α + 1)(β +1) = 1− c.
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 condition that the zeros of the polynomial f(x) = x3 + 3px2 + 3qx + r may be in A.P.
If the zeros of the polynomial f(x) = x3 − 12x2 + 39x + k are in A.P., find the value of k.
Find the zeroes of the quadratic polynomial `4x^2 - 4x + 1` and verify the relation between the zeroes and the coefficients.
Find the quadratic polynomial, sum of whose zeroes is `sqrt2` and their product is `(1/3)`.
On dividing `3x^3 + x^2 + 2x + 5` is divided by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively. Find g(x).
If 𝛼, 𝛽 are the zeroes of the polynomial f(x) = x2 + x – 2, then `(∝/β-∝/β)`
Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x),
where p(x) = x5 − 4x3 + x2 + 3x +1, g(x) = x3 − 3x + 1
Can the quadratic polynomial x2 + kx + k have equal zeroes for some odd integer k > 1?
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:
3x2 + 4x – 4
Find the sum and product of the roots of the quadratic equation 2x2 – 9x + 4 = 0.
Find the zeroes of the quadratic polynomial x2 + 6x + 8 and verify the relationship between the zeroes and the coefficients.
Find the zeroes of the polynomial x2 + 4x – 12.
