Question

Which of the following is a eighth-degree polynomial function? Select all that apply.
(1 point)
Responses

f left parenthesis x right parenthesis equals left parenthesis x superscript 4 baseline plus x right parenthesis squared
Image with alt text: f left parenthesis x right parenthesis equals left parenthesis x superscript 4 baseline plus x right parenthesis squared

f left parenthesis x right parenthesis equals Start Fraction 1 over x superscript 8 baseline minus 64 End Fraction
Image with alt text: f left parenthesis x right parenthesis equals Start Fraction 1 over x superscript 8 baseline minus 64 End Fraction

f left parenthesis x right parenthesis equals 10 minus x superscript 8 baseline
Image with alt text: f left parenthesis x right parenthesis equals 10 minus x superscript 8 baseline

f left parenthesis x right parenthesis equals x superscript 7 baseline plus 8 x minus 1

Answer 1

To identify the eighth-degree polynomial functions from the given options, we need to examine each function and determine its degree:

1. **\( f(x) = (x^4 + x)^2 \)**:
- This expression involves squaring a polynomial. The highest degree term inside the parentheses is \( x^4 \), and when squared, it becomes \( x^8 \). Thus, the overall degree of this polynomial is 8.
- This is an eighth-degree polynomial.

2. **\( f(x) = \frac{1}{x^8 - 64} \)**:
- This is not a polynomial function since polynomials cannot have negative powers of \( x \). The presence of \( x^{-8} \) indicates it is a rational function.
- This is not an eighth-degree polynomial.

3. **\( f(x) = 10 - x^8 \)**:
- This can be rewritten as \( f(x) = -x^8 + 10 \). The highest degree term here is \( -x^8 \), which means the degree of the polynomial is 8.
- This is an eighth-degree polynomial.

4. **\( f(x) = x^7 + 8x - 1 \)**:
- The highest degree term here is \( x^7 \), which means the degree of this polynomial is 7.
- This is not an eighth-degree polynomial.

Based on this analysis, the following functions are eighth-degree polynomial functions:

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

Thus, the answers are:

1. \( f(x) = (x^4 + x)^2 \)
2. \( f(x) = 10 - x^8 \)