Question

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

Answer 1

It is given that 2 and 0 are the zeroes of the polynomial $f\left(x\right)=2x^3-5x^2+ax+b$.
∴ f(2) = 0
$$\begin{gathered} \Rightarrow 2\times 2^3-5\times 2^2+a\times 2+b=0 \\ \Rightarrow 16-20+2a+b=0 \\ \Rightarrow -4+2a+b=0 \\ \Rightarrow 2a+b=4.....\left(1\right) \end{gathered}$$
Also,
f(0) = 0
$$\begin{gathered} \Rightarrow 2\times 0^3-5\times 0^2+a\times 0+b=0 \\ \Rightarrow 0-0+0+b=0 \\ \Rightarrow b=0 \end{gathered}$$
Putting b = 0 in (1), we get
$$\begin{gathered} 2a+0=4 \\ \Rightarrow 2a=4 \\ \Rightarrow a=2 \end{gathered}$$
Thus, the values of a and b are 2 and 0, respectively.