Advertisements
Advertisements
प्रश्न
If the zeros of the polynomial f(x) = ax3 + 3bx2 + 3cx + d are in A.P., prove that 2b3 − 3abc + a2d = 0.
उत्तर
Let a - d, a and a + d be the zeros of the polynomial f(x). Then,
Sum of the zeroes `=("coefficient of "x^2)/("coefficient of "x^3)`
`a-d+a+a+d=(-3b)/a`
`3a=(-3b)/a`
`a=(-3b)/axx1/3`
`a=(-b)/a`
Since a is a zero of the polynomial f(x).
Therefore,
f(x) = ax3 + 3bx2 + 3cx + d
f(a) = 0
f(a) = aa3 + 3ba2 + 3ca + d
aa3 + 3ba2 + 3ca + d = 0
`a((-b)/a)^3+3bxx((-b)/a)^2+3cxx((-b)/a)+d=0`
`axx(-b)/axx(-b)/axx(-b)/a+3xxbxx(-b)/axx(-b)/a+3xxcxx(-b)/a+d=0`
`(-b^3)/a^2+(3b^3)/a^2-(3cb)/a+d=0`
`(-b^3+3b^3-3abc+a^2d)/a^2=0`
2b3 − 3abc + a2d = 0 xa2
2b3 − 3abc + a2d = 0
Hence, it is proved that 2b3 − 3abc + a2d = 0
APPEARS IN
संबंधित प्रश्न
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
t2 – 15
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
`1/4 , -1`
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 the sum of the zeros of the quadratic polynomial f(t) = kt2 + 2t + 3k is equal to their product, find the value of k.
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 zeroes of the polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α − β)2.
Find the zeroes of the quadratic polynomial `2x^2 ˗ 11x + 15` and verify the relation between the zeroes and the coefficients.
What should be added to the polynomial x2 − 5x + 4, so that 3 is the zero of the resulting polynomial?
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 ______.
