Advertisements
Advertisements
Question
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
t2 – 15
Advertisements
Solution
h(t) `=t^2-15`
`=(t)^2-(sqrt15)^2`
`=(t+sqrt15)(t-sqrt15)`
For p(t) = 0, we have
Either `(t + sqrt15) = 0`
`t = sqrt15`
or `t - sqrt15 = 0`
`t = sqrt15`
Sum of the zeroes = `-("Coefficient of " t)/("Coefficient of " t^2)`
`-sqrt15+sqrt15=(-0)/1`
0 = 0
Also product of the zeroes = `"Constant term"/("Coefficient of "t^2)`
`-sqrt15xxsqrt15=(-15)/1`
`-15=-15`
Thus, the relationship between zeroes and the coefficients in the polynomial t2 - 15 is verified.
RELATED QUESTIONS
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
x2 – 2x – 8
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
3x2 – x – 4
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
1, 1
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
4, 1
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α2β + αβ2
If If α and β are the zeros of the quadratic polynomial f(x) = x2 – 2x + 3, find a polynomial whose roots are α + 2, β + 2.
Find the quadratic polynomial, sum of whose zeroes is `sqrt2` and their product is `(1/3)`.
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.
Find a cubic polynomial whose zeroes are 2, -3and 4.
If f(x) = `x^4– 5x + 6" is divided by g(x) "= 2 – x2`
Case Study -1
The figure given alongside shows the path of a diver, when she takes a jump from the diving board. Clearly it is a parabola.
Annie was standing on a diving board, 48 feet above the water level. She took a dive into the pool. Her height (in feet) above the water level at any time ‘t’ in seconds is given by the polynomial h(t) such that h(t) = -16t2 + 8t + k.
The zeroes of the polynomial r(t) = -12t2 + (k - 3)t + 48 are negative of each other. Then k 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?
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`4x^2 + 5sqrt(2)x - 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.
`(-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.
The zeroes of the quadratic polynomial x2 + 99x + 127 are ______.
If the zeroes of the polynomial x2 + px + q are double in value to the zeroes of the polynomial 2x2 – 5x – 3, then find the values of p and q.
If p(x) = x2 + 5x + 6, then p(– 2) is ______.
Find the zeroes of the polynomial x2 + 4x – 12.
The zeroes of the polynomial p(x) = 25x2 – 49 are ______.
