Question

A polynomial in x of degree 3 vanishes when x=1 and x=2, and has the values 4 and 28 when x=-1 and x=2, respectively. Then find the value of polynomial when x=0.

Answer 1

$f\left(x\right)=(x-1)(x+2)(ax+b)$

$f(-1)=4=(-2)\left(1\right)(b-a)$

$=a-b=2$

$f\left(2\right)=28=\left(1\right)\left(4\right)(2a+b)$

$2a+b=1$

solving $1$ and $2$ equations

$3a=9$

$a=3$

$b=1$

$\therefore f\left(x\right)=(x-1)(x+2)(3x+1)$

$f\left(0\right)=(-1)\left(2\right)\left(1\right)$

$=-2$