Question

If $2$ and $0$ are the zeroes of the polynomial $f\left(x\right)=2x^3-5x^2+ax+b$ then find the values of $a$ and $b$.

Answer 1

Given function is $f\left(x\right)=2x^3-5x^2+ax+b$

Since, if $k$ is a zero of polynomial $P\left(x\right)$ then $P\left(k\right)=0$.

It is given that, $2$ and $0$ are zeroes of the polynomial $f\left(x\right)$.

$\Rightarrow f\left(0\right)=0$

$\Rightarrow 2(0{)}^3-5(0{)}^2+a\left(0\right)+b=0$

$\Rightarrow b=0$---(1)

and,

$\Rightarrow f\left(2\right)=0$

$\Rightarrow 2(2{)}^3-5(2{)}^2+a\left(2\right)+b=0$

$\Rightarrow 16-20+2a+0=0$ [$\because b=0$]

$\Rightarrow 2a-4=0$

$\Rightarrow 2a=4$

$\Rightarrow a=\frac{4}{2}$

$\Rightarrow a=2$ ---(2)

Hence, $a=2,b=0$.