Find the condition which must be sa...
Question

# Find the condition which must be satisfied by the coefficients of the polynomial f(x)=x3-(x2+ mx-n. Given that the sum of the two zeroes is zero.

IIT/JEE
Maths
Solution
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Let ( alpha, beta ) and ( gamma ) be the zeroes of the polynomial ( f(x) ) such that ( alpha+beta=0 ) Now, ( alpha+beta+gamma=frac{text { Coefficient of } x^{2}}{text { Coefficient of } x^{3}} ) since, ( gamma ) is a zero of the polynomial ( f(x) ). Therefore, ( f(gamma)=0 ) ( Rightarrow gamma^{3}-ell gamma^{2}+m gamma-n=0 quad[because gamma=ell] ) ( Rightarrow ell^{3}-ell^{3}+mathrm{m} ell-mathrm{n}=0 Rightarrow mathrm{m} ell=mathrm{n} ) This is the required condition.