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

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

$3=a(1)(5)$

$3=5a$

$\dfrac{3}{5}=a$

Putting $a=\dfrac{3}{5}$ in (i), we get

$P(x)=\dfrac{3}{5}(x-1)(x+3)$

Therefore, the required polynomial is $P(x)=\dfrac{3}{5}(x-1)(x+3)$.