Question

what's the formula for finding the cubic polynomial whose zeros r given to you. Explain with the help of an example

Answer 1

hi friend,

if p,q,r be the zeros of the polynomial given..

then we need to find p+q+r ,pq+qr+rs,pqr

then the cubic polynomial will be

x³-(p+q+r)x²+(pq+qr+rs)x-pqr

example :- let 1,1,1 be the zeros

then p+q+r=1+1+1=3

pq+qr+rs=1+1+1=3

pqr=1

so the required cubic polynomial is

x³-3x²+3x-1


I hope this will help u ;)

Answer 2

Let a,b,c are the zeroes of the given polynomial.

The cubic polynomial is in the form of x³-(a+b+c)x²+(ab+bc+ca)x-abc

So,here when zeroes of the polynomial are given,first and must we need to find sum of zeroes,sum of the product of each two zeroes and product of all zeroes.

a+b+c = -x² coefficient/x³ coefficient

ab+bc+ca=x coefficient/x³ coefficient

abc= -constant/x³ coefficient

By this relation between zeroes and coefficients of the given polynomial, we can find the values of a+b+c and ab+bc+ca and abc.

Then we need to substitute those values in the cubic polynomial,
x³-(a+b+c)x²+(ab+bc+ca)x-abc

Thus,we get the required cubic polynomial.

For example,

Let's take 1,2,3 as the zeroes of the cubic polynomial.

a=1,b=2,c=3

a+b+c=1+2+3=6/1 =-(-6)/1= -x² coefficient/x³ coefficient

ab+bc+ca=(1×2)+(2×3)+(1×3)=(2+6+3)=11/1= x coefficient/x³ coefficient

abc=(1)(2)(3)=6/1=-(-6)/1 = -constant/x³ coefficient

x³-6x²+11x+6 is the required cubic polynomial.

We can also find like this,

x³ coefficient=1
x² coefficient= -6
x coefficient = 11
Constant= -6

Cubic polynomial is
1x³+(-6)x²+11x+(-6)
x³-6x²+11x-6

Hope you understand