Question

How do you find the polynomial function with leading coefficient $2$ that has the given degree and zeroes: degree $3$:zeroes $-2,1,4?$

Answer 1

Find the polynomial function with leading coefficient $2$ that has the degree $3$:

It is given the zeroes of polynomial function are $-2,1$ and $4$.

Let the polynomial function be $p\left(x\right)$:

If $\text{'}a\text{'}$ is the zero of any polynomial function, then $(x-a)$ will be root of that polynomial function.

$(x+2)(x-1)(x-4)$ as root of $p\left(x\right)$:

Since the leading coefficient of $p\left(x\right)$ is $2$, $p\left(x\right)$ will be as follows:

$p\left(x\right)=2(x+2)(x-1)(x-4)$

$\begin{array}{rcl}& \Rightarrow & p\left(x\right)=2(x^3-3x^2-6x+8)\\ & =& 2x^3-6x^2-12x+16\end{array}$

Hence, the required polynomial function is $\mathbf{2}{\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{6}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{12}\mathit{x}\mathbf{+}\mathbf{16}\mathbf{.}$