Question

which of the following is a eighth - degree polynomial function? select all that apply.
(1 point)
□ $f(x)=(x^{4}+x)^{2}$
□ $f(x)=\frac{1}{x^{8}-64}$
□ $f(x)=10x - x^{8}$
□ $f(x)=x^{7}+8x - 1$

Explanation:

Step1: Recall the definition of a polynomial function and degree of a polynomial

A polynomial function is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non - negative integer exponents of variables. The degree of a polynomial is the highest power of the variable in the polynomial.

Step2: Analyze \(f(x)=(x^{4}+x)^{2}\)

Expand \((x^{4}+x)^{2}\) using the formula \((a + b)^{2}=a^{2}+2ab + b^{2}\), where \(a = x^{4}\) and \(b=x\).
\((x^{4}+x)^{2}=(x^{4})^{2}+2\times x^{4}\times x+x^{2}=x^{8}+2x^{5}+x^{2}\). The highest power of \(x\) is \(8\), and it is a polynomial.

Step3: Analyze \(f(x)=\frac{1}{x^{8}-64}\)

This function is a rational function (a ratio of two polynomials), not a polynomial function because the variable \(x\) is in the denominator. So it is not a polynomial.

Step4: Analyze \(f(x)=10x - x^{8}\)

This is a polynomial function. The highest power of \(x\) is \(8\) (from the term \(-x^{8}\)).

Step5: Analyze \(f(x)=x^{7}+8x - 1\)

This is a polynomial function. The highest power of \(x\) is \(7\) (from the term \(x^{7}\)).

Answer:

• \(f(x)=(x^{4}+x)^{2}\)
• \(f(x)=10x - x^{8}\)