Question

Verify whether $-1$ and $\frac{3}{2}$ are zeros of the polynomial $p\left(x\right)=2x^3-9x^2+x+12$.

Answer 1

Step 1: Determine whether $-1$ is the zeros of the given polynomial $p\left(x\right)=2x^3-9x^2+x+12$.

A real number $c$ is called a zero of the polynomial $p\left(x\right)$ if $p\left(c\right)=0$.

$$\begin{gathered} p(-1)=0 \\ \Rightarrow 2{\left(-1\right)}^3-9{\left(-1\right)}^2+\left(-1\right)+12=0 \\ \Rightarrow -2-9-1+12=0 \\ \Rightarrow 0=0 \end{gathered}$$

Therefore, $-1$ is the zeros of the polynomial $p\left(x\right)=2x^3-9x^2+x+12$.

Step 2: Determine whether $\frac{3}{2}$ is the zeros of the given polynomial $p\left(x\right)=2x^3-9x^2+x+12$.

$$\begin{gathered} p\left(\frac{3}{2}\right)=0 \\ \Rightarrow 2{\left(\frac{3}{2}\right)}^3-9{\left(\frac{3}{2}\right)}^2+\frac{3}{2}+12=0 \\ \Rightarrow \frac{27}{4}-\frac{81}{4}+\frac{3}{2}+12=0 \\ \Rightarrow \frac{27-81+6+48}{4}=0 \\ \Rightarrow 0=0 \end{gathered}$$

Hence, it is verified that $-1$ and $\frac{3}{2}$ are zeros of the polynomial $p\left(x\right)=2x^3-9x^2+x+12$.