Which of the following is a eighth-degree polynomial function? Select all that apply.

Question

(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:

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