Answer
The given function \[g\left( x \right)=6{{x}^{7}}+\pi {{x}^{5}}+\frac{2}{3}x\]is a polynomial with degree 7.
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.
Now, the function g(x) is called a polynomial function of degree n; its coefficient is called the leading coefficient.
Now, the given function $g\left( x \right)=6{{x}^{7}}+\pi {{x}^{5}}+\frac{2}{3}x$ satisfies all the conditions to be a polynomial even though the coefficient of ${{x}^{5}}$ is $\pi $ , which is irrational. The definition of 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 7 then the degree of the polynomial is 7.
Hence, the given function $g\left( x \right)=6{{x}^{7}}+\pi {{x}^{5}}+\frac{2}{3}x$ is a polynomial with degree 7.
