Advertisements
Advertisements
Question
If If α and β are the zeros of the quadratic polynomial f(x) = x2 – 2x + 3, find a polynomial whose roots are `(alpha-1)/(alpha+1)` , `(beta-1)/(beta+1)`
Advertisements
Solution
Since α and β are the zeros of the quadratic polynomial f(x) = x2 – 2x + 3
`alpha+beta="-coefficient of x"/("coefficient of "x^2)`
`=(-(-2))/1`
= 2
Product of the zeroes `="constant term"/("coefficient of "x^2)`
`=3/1`
= 3
Let S and P denote respectively the sums and product of the polynomial whose zeros
`(alpha-1)/(alpha+1)` , `(beta-1)/(beta+1)`
`S=(alpha-1)/(alpha+1)+(beta-1)/(beta+1)`
`S=((alpha-1)(beta+1)(beta-1)(alpha+1))/((alpha+1)(beta+1))`
`S=(alphabeta-beta+alpha-1+alphabeta-alpha+beta-1)/(alphabeta+beta+alpha+1)`
`S=(alphabeta+alphabeta-1-1)/(alphabeta+(alpha+beta)+1)`
By substituting α + β = 2 and αβ = 3 we get,
`S=(3+3-1-1)/(3+2+1)`
`S=(6-2)/6`
`S=4/6`
`P=((alpha-1)/(alpha+1))((beta-1)/(beta+1))`
`P=(alphabeta-beta-alpha+1)/(alphabeta+beta+alpha+1)`
`P=(alphabeta-(beta+alpha)+1)/(alphabeta+(alpha+beta)+1)`
`P=(3-2+1)/(3+2+1)`
`P=2/6`
`P=1/3`
The required polynomial f (x) is given by,
f(x) = k(x2 - Sx + P)
`f(x) = k(x^2 - 2/3x + 1/3)`
Hence, the required equation is `f(x) = k(x^2 - 2/3x + 1/3)` , where k is any non zero real number .
APPEARS IN
RELATED QUESTIONS
Find the zeros of the quadratic polynomial 4x2 - 9 and verify the relation between the zeros and its coffiecents.
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
x2 – 2x – 8
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
1, 1
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(y) = 5y2 − 7y + 1, find the value of `1/alpha+1/beta`
If one zero of the quadratic polynomial f(x) = 4x2 − 8kx − 9 is negative of the other, find the value of k.
If α and β are the zeros of the quadratic polynomial f(x) = x2 − 3x − 2, find a quadratic polynomial whose zeroes are `1/(2alpha+beta)+1/(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.
Find the zeroes of the quadratic polynomial `(8x^2 ˗ 4)` 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.
If `x =2/3` and x = -3 are the roots of the quadratic equation `ax^2+2ax+5x ` then find the value of a and b.
By actual division, show that x2 – 3 is a factor of` 2x^4 + 3x^3 – 2x^2 – 9x – 12.`
If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are each equal to zero, then the third zero is
Basketball and soccer are played with a spherical ball. Even though an athlete dribbles the ball in both sports, a basketball player uses his hands and a soccer player uses his feet. Usually, soccer is played outdoors on a large field and basketball is played indoor on a court made out of wood. The projectile (path traced) of soccer ball and basketball are in the form of parabola representing quadratic polynomial.
What will be the expression of the polynomial?
If all three zeroes of a cubic polynomial x3 + ax2 – bx + c are positive, then at least one of a, b and c is non-negative.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`2x^2 + (7/2)x + 3/4`
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.
A quadratic polynomial whose sum and product of zeroes are 2 and – 1 respectively is ______.
