Chapter 4.1, Problem 17AYU

In Problems $15-26$, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, state why not. Write each polynomial in standard form. Then identify the leading term and the constant term.

$g\left(x\right)=\frac{2+3x^2}{5}$

To determine

Whether the function $g\left(x\right)=\frac{2+3x^2}{5}$ is a polynomial or not. If it is polynomial, state the degree and for those that are not, state why not. Write a polynomial in standard form, identify the leading term and the constant term.

Answer to Problem 17AYU

Solution:

The function $g\left(x\right)=\frac{2+3x^2}{5}$ is a polynomial of degree 2.

The standard form of polynomial is $\frac{3}{5}{x}^2+0x+\frac{2}{5}$ with leading term $\frac{3}{5}$ and constant term $\frac{2}{5}$.

Explanation of Solution

Given Information:

The function $g\left(x\right)=\frac{2+3x^2}{5}$

A polynomial function in one variable is a function of the form

$f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...........}+a_{1}x+a_0$

Where $a_n,a_{n-1},\mathrm{.....},a_1,a_0$ are constants, called the coefficients of the polynomial, $n\ge 0$ is an integer and $x$ is a variable. If $a_n\ne 0$, it is called the leading coefficient and n is the degree of polynomial, $a_0$ is called the constant term.

The function $g\left(x\right)=\frac{2+3x^2}{5}$ can be written as $g\left(x\right)=\frac{2}{5}+\frac{3}{5}{x}^2$

That is, $g\left(x\right)=\frac{2}{5}+0x+\frac{3}{5}{x}^2$

Hence, the function $g\left(x\right)=\frac{2+3x^2}{5}$ is a polynomial in variable $x$, and $\frac{2}{5},0,\frac{3}{5}$ are coefficients of the polynomial. All powers of $x$ are non-negative integers.

The highest power of $x$ in the function $g\left(x\right)=\frac{2}{5}+0x+\frac{3}{5}{x}^2$ is 2, so the degree is 2.

To write the polynomial in standard form, begin with a non-zero term of highest degree and according to the degree, continue with terms in descending order.

Hence, the polynomial in standard form is written as $\frac{3}{5}{x}^2+0x+\frac{2}{5}$

For this polynomial the leading term is $\frac{3}{5}$ and constant term is $\frac{2}{5}$.

Therefore, the function $g\left(x\right)=\frac{2+3x^2}{5}$ is a polynomial of degree 2 with leading term $\frac{3}{5}$ and constant term $\frac{2}{5}$