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: Analyze \( f(x) = (x^4 + \pi)^2 \)

Expand using \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = x^4\), \(b=\pi\). So \(f(x)=x^8 + 2\pi x^4+\pi^2\). The highest power of \(x\) is 8, so it's an 8th - degree polynomial.

Step2: Analyze \( f(x)=\frac{1}{x^8 - 64}\)

This is a rational function (a fraction of two polynomials), not a polynomial function because the variable \(x\) is in the denominator. So it's not a polynomial, let alone an 8th - degree polynomial.

Step3: Analyze \( f(x)=10x - x^8\)

The highest power of \(x\) in this polynomial is 8 (from the term \(-x^8\)), so it is an 8th - degree polynomial.

Step4: Analyze \( f(x)=x^7 + 8x - 1\)

The highest power of \(x\) here is 7, so it is a 7th - degree polynomial, not an 8th - degree polynomial.

Answer:

The eighth - degree polynomial functions are \(f(x)=(x^4 + \pi)^2\) and \(f(x)=10x - x^8\), so the correct options are:

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