Advertisements
Advertisements
Questions
If α, β are the zeroes of the polynomial f(x) = x2 + x – 2, then `(α/β - α/β)`.
If α and β are the zeros of the polynomial f(x) = x2 + x – 2, find the value of `(1/α - 1/β)`.
Advertisements
Solution
By using the relationship between the zeroes of the quadratic polynomial.
We have
Sum of zeroes = `(-("Coefficient of" x))/("Coefficient of" x^2)` and Product of zeroes = `("Constant term")/("Coefficient of" x^2)`
∴ `α + β = (-1)/1` and `αβ = (-2)/1`
⇒ α + β = –1 and αβ = –2
Now, `(1/α - 1/β)^2 = ((β - α)/(αβ))^2`
= `((α + β)^2 - 4αβ)/(αβ)^2` ...[∵ (β – α)2 = (α + β)2 – 4αβ]
= `((-1)^2 - 4(-2))/((-2)^2)` ...[∵ α + β = –1 and αβ = –2]
= `((-1)^2 - 4(-2))/4`
= `9/4`
∵ `(1/α - 1/β)^2 = 9/4`
⇒ `1/α - 1/β = +-3/2`
APPEARS IN
RELATED QUESTIONS
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.
6x2 – 3 – 7x
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
3x2 – x – 4
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)`
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 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)`
Find the zeroes of the polynomial f(x) = `2sqrt(3)x^2 - 5x + sqrt(3)` and verify the relation between its zeroes and coefficients.
Find the zeroes of the quadratic polynomial 4x2 – 4x + 1 and verify the relation between the zeroes and the coefficients.
Find a cubic polynomial whose zeroes are `1/2`, 1 and –3.
Find the quotient and the remainder when f(x) = x4 – 5x + 6 is divided by g(x) = 2 – x2.
What should be added to the polynomial x2 − 5x + 4, so that 3 is the zero of the resulting polynomial?
An asana is a body posture, originally and still a general term for a sitting meditation pose, and later extended in hatha yoga and modern yoga as exercise, to any type of pose or position, adding reclining, standing, inverted, twisting, and balancing poses. In the figure, one can observe that poses can be related to representation of quadratic polynomial.
The zeroes of the quadratic polynomial `4sqrt3"x"^2 + 5"x" - 2sqrt3` are:
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 the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all 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:
t3 – 2t2 – 15t
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`
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.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`v^2 + 4sqrt(3)v - 15`
If one zero of the polynomial p(x) = 6x2 + 37x – (k – 2) is reciprocal of the other, then find the value of k.
If α, β are zeroes of quadratic polynomial 5x2 + 5x + 1, find the value of α–1 + β–1.
