Advertisements
Advertisements
प्रश्न
If the zeros of the polynomial f(x) = 2x3 − 15x2 + 37x − 30 are in A.P., find them.
Advertisements
उत्तर
Let α = a - d, β = a and γ = a + d be the zeros of the polynomial
f(x) = 2x3 − 15x2 + 37x − 30
Therefore
`alpha+beta+gamma=("coefficient of "x^2)/("coefficient of "x^3)`
`=-((-15)/2)`
`=15/2`
`alphabetagamma="-constant term"/("coefficient of "x^2)`
`=-((-30)/2)`
= 15
Sum of the zeroes `=("coefficient of "x^2)/("coefficient of "x^3)`
`(a-d)+a+(a+d)=15/2`
`a+a+a-d+d=15/2`
`3a=15/2`
`a=15/2xx1/3`
`a=5/2`
Product of the zeroes `="-constant term"/("coefficient of "x^2)`
`alphabetagamma=15`
`(a-d)+a+(a+d)=15`
`a(a^2-d^2)=15`
Substituting a = 5/2 we get
`5/2((5/2)^2-d^2)=15`
`5/2(25/4-d^2)=15`
`25/4-d^2=15xx2/5`
`25/4-d^2=3xx2`
`25/4-d^2=6`
`-d^2=6-25/4`
`-d^2=(24-25)/4`
`-d^2=(-1)/4`
`d^2=1/4`
`d xx d=1/2xx1/2`
`d=1/2`
Therefore, substituting a=5/2 and d=1/2 in α = a - d, β = a and γ = a + d
α = a - d
`alpha=5/2-1/2`
`alpha=(5-1)/2`
`alpha=4/2`
`alpha=2`
β = a
`beta=5/2`
γ = a + d
`gamma=5/2+1/2`
`gamma=(5+1)/2`
`gamma=6/2`
`gamma=3`
Hence, the zeros of the polynomial are 2, `5/2` , 3.
संबंधित प्रश्न
if α and β are the zeros of ax2 + bx + c, a ≠ 0 then verify the relation between zeros and its cofficients
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 α and β are the zeros of the quadratic polynomial f(t) = t2 − 4t + 3, find the value of α4β3 + α3β4.
If α and β are the zeros of the quadratic polynomial f(x) = x2 − 1, find a quadratic polynomial whose zeroes are `(2alpha)/beta" and "(2beta)/alpha`
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 a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and product of its zeros as 3, −1 and −3 respectively.
Find a cubic polynomial whose zeroes are `1/2, 1 and -3.`
If f(x) =` x^4 – 3x^2 + 4x + 5` is divided by g(x)= `x^2 – x + 1`
If 3 and –3 are two zeroes of the polynomial `(x^4 + x^3 – 11x^2 – 9x + 18)`, find all the zeroes of the given polynomial.
A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is
If x + 2 is a factor of x2 + ax + 2b and a + b = 4, then
If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
5t2 + 12t + 7
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`
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.
`-2sqrt(3), -9`
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 the zeroes of the polynomial p(x) = 4x2 – 3x – 7, then `(1/α + 1/β)` is equal to ______.
