Question

if [x-a} is a factor of the polynomials [x^2+px-q] and [x^2+rx-t] prove that a=t-q/r-p

Answer 1

Given :

(x+a) is a factor of x

2

+px+q and x

2

+mx+n

then using the factor theorem which says that the polynomial f(x0 has a factor (x−k) if and only if f(k)=0

We have

(−a)

2

+p(−a)+q=0⟶(1)

⇒a

2

−ap+q=0⟶(2)

and

(−a)

2

+m(−a)+n=0⟶(3)

⇒a

2

−ma+n=0⟶(4)

Subtracting (2) & (4) we get

−ap+am+q−n=0

⇒+a(m−p)=n−q

⇒a=

m−p

n−q

Hence, proved