Answer
The function given \[g\left( x \right)=7{{x}^{5}}-\pi {{x}^{3}}+\frac{1}{5}x\]is a polynomial with degree 5.
Work Step by Step
A function is said to be a polynomial function if
$g\left( x \right)={{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+\cdots +{{a}_{2}}{{x}^{2}}+{{a}_{1}}x+{{a}_{0}}$ ,
Where ${{a}_{n}},{{a}_{n-1}},\ldots ,{{a}_{2}},{{a}_{1}},{{a}_{0}}$ are any real numbers with ${{a}_{n}}\ne 0$ and n is any non-negative integer. The function g(x) is called a polynomial function of degree n, its coefficient is called the leading coefficient.
Now, the given function $g\left( x \right)=7{{x}^{5}}-\pi {{x}^{3}}+\frac{1}{5}x$ satisfies all the conditions to be a polynomial even though the coefficient of ${{x}^{3}}$ is $-\pi $ , which is irrational. The definition of a polynomial states that the coefficient of the variables should be real; hence, this function is a polynomial and since the highest power of f(x) is 5 then the degree of the polynomial is 5.
Hence, the function $g\left( x \right)=7{{x}^{5}}-\pi {{x}^{3}}+\frac{1}{5}x$ is a polynomial function with degree $5$.
