Advertisements
Advertisements
Question
Find the condition that the zeros of the polynomial f(x) = x3 + 3px2 + 3qx + r may be in A.P.
Advertisements
Solution
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=(-3p)/1`
`3a=-3p`
`a=(-3xxp)/3`
a = -p
Since 'a' is a zero of the polynomial f(x). Therefore,
f(x) = x3 + 3px2 + 3qx + r
f(a) = 0
f(a) = a3 + 3pa2 + 3qa + r
a3 + 3pa2 + 3qa + r = 0
Substituting a = -p we get,
(-p)3 + 3p(-p)2 + 3q(-p) + r = 0
-p3 + 3p3 - 3pq + r = 0
2p3 - 3pq + r = 0
Hence, the condition for the given polynomial is 2p3 - 3pq + r = 0
APPEARS IN
RELATED QUESTIONS
Find the zeros of the quadratic polynomial 9x2 - 5 and verify the relation between the zeros and its coefficients.
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
t2 – 15
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes, respectively.
`sqrt2 , 1/3`
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case
x3 – 4x2 + 5x – 2; 2, 1, 1
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
`q(x)=sqrt3x^2+10x+7sqrt3`
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α - β
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α4 + β4
If α and β are the zeroes of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/(aalpha+b)+1/(abeta+b)`.
If a and are the zeros of the quadratic polynomial f(x) = ЁЭСе2 − ЁЭСе − 4, find the value of `1/alpha+1/beta-alphabeta`
If α and β are the zeros of a quadratic polynomial such that α + β = 24 and α − β = 8, find a quadratic polynomial having α and β as its zeros.
If the zeros of the polynomial f(x) = 2x3 − 15x2 + 37x − 30 are in A.P., find them.
Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.
Find the quadratic polynomial, sum of whose zeroes is 0 and their product is -1. Hence, find the zeroes of the polynomial.
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.
If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are each equal to zero, then the third zero is
The polynomial which when divided by −x2 + x − 1 gives a quotient x − 2 and remainder 3, is
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`7y^2 - 11/3 y - 2/3`
A quadratic polynomial the sum and product of whose zeroes are – 3 and 2 respectively, is ______.
A quadratic polynomial whose sum and product of zeroes are 2 and – 1 respectively is ______.
