Chapter 4.1, Problem 25AYU

In Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term.

$G\left(x\right)=2{\left(x-l\right)}^2(x^2+1)$

To determine

The function is polynomial.

The degree of the polynomial.

Polynomial in standard form.

Leading term and constant term.

Answer to Problem 25AYU

$G(x)=2{(x-1)}^2(x^2+1)\text{ is a polynomial since the highest power on }x=\text{ degree }=4\text{ which is nonnegative}$.

$\text{Degree of the polynomial }=\text{ 4}$.

$\text{Polynomial in standard form }=G(x)=2x^4-4x^3+4x^2-4x+1$.

$\text{Leading term }=2\text{ and constant term }=1$.

Explanation of Solution

Given:

$G\left(x\right)=2{\left(x-1\right)}^2\left(x^2+1\right)$

$G\left(x\right)=2{\left(x-1\right)}^2\left(x^2+1\right)=2\left(x^2-2x+1\right)\left(x^2+1\right)=2x^4-4x^3+4x^2-4x+1$ is a polynomial since the highest power on $x=\text{ degree }=4$ which is nonnegative.

Degree of the $\text{polynomial }=4$.

Polynomial in standard $\text{form }=G\left(x\right)=2x^4-4x^3+4x^2-4x+1$.

Leading $\text{term }=\text{ coefficient}$ of $x^4=2$ and constant $\text{term }=\text{ term}$ without $\text{variable }=\text{ 1}$.