Question

16.Find the condition that zeroes of the polynomial p(x)= - 3px²+qx - r are in arithmetic progession.
Ans: 2p - pq-r
17. Find the zeros of the polynomial and verify the relation between the
zeros and the coefficients of the polynomial :q(s) – 2s² + 5s + 3​

Answer 1

Answer:

Let α,β and ψ be the zeros of the polynomial f(x)=x

2

−3px

2

+qx−r

f(x)=x

2

−3px

2

+qx−r

=(x−α)(x−β)(x−ψ)

=x

2

−(α+β+ψ)x

2

+(αβ+βψ+ψα)x−αβψ

Equating the coefficients of x

2

we have

−(α+β+ψ)=−3p

α+β+ψ=3p ……….(1)

Now, it is given that α,β & ψ are in A.P. Let S be the common difference of terms of the AP.

⇒β−α=δ

⇒α=β−δ ………….(2)

ψ−β=δ

⇒ψ=β+δ ……………..(3)

Put the values of α and ψ from equation (2) and (3) in (1)

β−δ+β+β+δ=3p

⇒3β=3p

⇒β=p

⇒ p is a root of the polynomial f(x)

put x=p in f(x)

⇒0=f(p)=(p)

3

−3p(p)

2

+q(p)−r

p

3

−3p

3

+qp−r=0

−2p

2

+qp−r=0

⇒2p

3

=qp−r.