Advertisements
Advertisements
Question
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.
Advertisements
Solution
Since α and β are the zeros of the quadratic polynomial f(x) = x2 − p (x + 1) — c
Then
x2 - p(x + 1) - c
x2 - px - p - c
`alpha+beta="-coefficient of x"/("coefficient of "x^2)`
`=(-(-p))/1`
= p
`alphabeta="constant term"/("coefficient of "x^2)`
`=(-p-c)/1`
= -p-c
We have to prove that (α + 1)(β +1) = 1 − c
(α + 1)(β +1) = 1 - c
(α + 1)β + (α +1)(1) = 1 - c
αβ + β + α + 1 = 1 - c
αβ + (α + β) + 1 = 1 - c
Substituting α + β = p and αβ = -p-c we get,
-p - c + p + 1 = 1 - c
1 - c = 1 - c
Hence, it is shown that (α + 1)(β +1) = 1 - c
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:
3x2 – x – 4
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) = x2 − 5x + 4, find the value of `1/alpha+1/beta-2alphabeta`
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 a cubic polynomial whose zeroes are `1/2`, 1 and –3.
If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, the\[\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} =\]
If α, β are the zeros of the polynomial f(x) = ax2 + bx + c, then\[\frac{1}{\alpha^2} + \frac{1}{\beta^2} =\]
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 ______.
The below picture are few natural examples of parabolic shape which is represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.
If the sum of the roots is –p and the product of the roots is `-1/"p"`, then the quadratic polynomial is:
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:
The number of polynomials having zeroes as –2 and 5 is ______.
If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is –1, then the product of the other two zeroes is ______.
If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
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.
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.
`(-3)/(2sqrt(5)), -1/2`
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`y^2 + 3/2 sqrt(5)y - 5`
If α, β are the zeroes of the polynomial p(x) = 4x2 – 3x – 7, then `(1/α + 1/β)` is equal to ______.
A quadratic polynomial the sum and product of whose zeroes are – 3 and 2 respectively, is ______.
Find the zeroes of the polynomial x2 + 4x – 12.
