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: Define 8th-degree polynomial

An 8th-degree polynomial is a function where the highest power of $x$ is 8, with only non-negative integer exponents on $x$, and no variables in denominators.

Step2: Analyze $f(x)=(x^4+x)^2$

Expand the function:
$$(x^4+x)^2 = x^8 + 2x^5 + x^2$$
The highest power of $x$ is 8, so it is an 8th-degree polynomial.

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

This is a rational function (variable in denominator, equivalent to $(x^8-64)^{-1}$), so it is not a polynomial.

Step4: Analyze $f(x)=10x - x^8$

Rewrite as $f(x)=-x^8 + 10x$. The highest power of $x$ is 8, so it is an 8th-degree polynomial.

Step5: Analyze $f(x)=x^7+8x-1$

The highest power of $x$ is 7, so it is a 7th-degree polynomial, not 8th-degree.

Answer:

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