Question

Find a
polynomial P(x) of 2nd degree if P(1)=0
P (2) 3
P(-3)=0​

Answer 1

Given:

P(x) is a 2nd degree polynomial.

`P(1)=0,\ P(2)=3,\ P(-3)=0`

To find:

The polynomial P(x).

Solution:

If P(x) is a polynomial and P(c)=0, then c is a zero of the polynomial and (x-c) is a factor of polynomial P(x).

We have,
`P(1)=0,\ P(-3)=0`. It means 1 and -3 are two zeros of the polynomial P(x) and (x-1) and (x+3) are two factors of the polynomial P(x).

So, the required polynomial is defined as:

`P(x)=a(x-1)(x+3)` ...(i)

Where, a is a constant.

We have,
`P(2)=3`. So, substituting
`x=2,\ P(x)=3` in (i), we get

`3=a(2-1)(2+3)`

`3=a(1)(5)`

`3=5a`

`(3)/(5)=a`

Putting
`a=(3)/(5)` in (i), we get

`P(x)=(3)/(5)(x-1)(x+3)`

Therefore, the required polynomial is
`P(x)=(3)/(5)(x-1)(x+3)`.