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Chapter 3: Problem 4

Determine which functions are polynomial functions. For those that are, identify the degree. $$g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x$$

Short Answer

Expert verified
The function \(g(x) = 6x^{7}+\pi x^{5}+\frac{2}{3} x\) is a polynomial function and its degree is 7.

Step by step solution

01

Determine if the function is a polynomial

A function is a polynomial if it is defined for all real numbers, the powers on the variable are whole numbers, and the coefficients are real numbers. In the given function \(g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x\), \(x\) has only whole number exponents (7, 5 and 1) and the coefficients 6, \(\pi\) and \(\frac{2}{3}\) are all real numbers. Therefore, \(g(x)\) is a polynomial function.
02

Identify the degree of the polynomial

In a polynomial function, the degree is the highest power of the variable. For the function \(g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x\), the highest power of \(x\) is 7. Hence, the degree of the polynomial \(g(x)\) is 7.

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