Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
`0, sqrt5`
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
If the zeroes of the polynomial x3 – 3x2 + x + 1 are a – b, a, a + b, find a and b
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
`p(x) = x^2 + 2sqrt2x + 6`
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
`f(x)=x^2-(sqrt3+1)x+sqrt3`
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α2β + αβ2
If 𝛼 and 𝛽 are the zeros of the quadratic polynomial p(x) = 4x2 − 5x −1, find the value of α2β + αβ2.
If the squared difference of the zeros of the quadratic polynomial f(x) = x2 + px + 45 is equal to 144, find the value of p.
Verify that 5, -2 and 13 are the zeroes of the cubic polynomial `p(x) = (3x^3 – 10x^2 – 27x + 10)` and verify the relation between its zeroes and coefficients.
If f(x) =` x^4 – 3x^2 + 4x + 5` is divided by g(x)= `x^2 – x + 1`
By actual division, show that x2 – 3 is a factor of` 2x^4 + 3x^3 – 2x^2 – 9x – 12.`
Define a polynomial with real coefficients.
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?
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:
3x2 + 4x – 4
If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it ______.
If α, β are the zeroes of the polynomial p(x) = 4x2 – 3x – 7, then `(1/α + 1/β)` is equal to ______.
Find the zeroes of the polynomial x2 + 4x – 12.
The zeroes of the polynomial p(x) = 2x2 – x – 3 are ______.
