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
Image with alt text: f left parenthesis x right parenthesis equals x superscript 7 baseline plus 8 x minus 1

Answer 1

To determine which of the given functions is an eighth-degree polynomial function, we need to analyze each option:

1. **\( f(x) = (x^4 + x)^2 \)**:
- This function combines terms and raises them to the second power.
- The highest degree term when expanded would be \( x^8 \) (since \( (x^4)^2 = x^8 \)).
- Therefore, this **is an eighth-degree polynomial function**.

2. **\( f(x) = \frac{1}{x^8 - 64} \)**:
- This function is a rational function, not a polynomial, since it includes a division by \( x^8 - 64 \).
- Thus, this **is not an eighth-degree polynomial function**.

3. **\( f(x) = 10 - x^8 \)**:
- This function can be rearranged to \( -x^8 + 10 \), with \( -x^8 \) as the leading term.
- The highest degree term is \( x^8 \).
- Therefore, this **is an eighth-degree polynomial function**.

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

### Summary:
The eighth-degree polynomial functions among the options are:
- \( f(x) = (x^4 + x)^2 \)
- \( f(x) = 10 - x^8 \)