Question

If $0$ and $1$ are the zeroes of the polynomial $f\left(x\right)=2x^3-3x^2+ax+b$, find the values of $a$ and $b$

Answer 1

Given polynomial $f\left(x\right)=2x^3-3x^2+ax+b$ is zero if x is 0 and 1

Replace x by 0 we get

$2(0{)}^3-3(0{)}^2+a\left(0\right)+b=0$

$\Rightarrow 0-0+0+b=0$

$\Rightarrow b=0$

Replace x by 1 we get

$2(1{)}^3-3(1{)}^2+a\left(1\right)+b=0$

$\Rightarrow 2\times 1-3\times 1+a\times 1+b=0$

$\Rightarrow 2-3+a+b=0$

$\Rightarrow -1+a+0=0$ (Put the value of b=0)

$\Rightarrow -1+a=0$

Add 1 both side we get

$-1+a+1=0+1$

$\Rightarrow a=1$

Then $a=1$ and $b=0$