Question

using structure write the function below in standard form.
$f(x) = 5x^3x + \frac{5}{2}x^3 - 9x^4 + \sqrt{2}x^2 + 4x - 1 - x^{-5}x^5 - 4$
$f(x) = \square$
the degree is $\boldsymbol{\vee}$.
the function is a $\boldsymbol{\vee}$ function.
the leading coefficient is $\boldsymbol{\vee}$.

Explanation:

Step1: Simplify like variable terms

$5x^3x = 5x^{3+1}=5x^4$, $x^{-5}x^5=x^{-5+5}=x^0=1$

Step2: Combine like polynomial terms

Combine $x^4$ terms: $5x^4 - 9x^4 = -4x^4$; combine constants: $-1 - 1 - 4 = -6$

Step3: Write in standard form

Arrange terms by descending exponents:
$f(x)=-4x^4+\frac{5}{2}x^3+\sqrt{2}x^2+4x-6$

Step4: Identify degree

Highest exponent is 4

Step5: Classify function type

Degree 4 means quartic polynomial

Step6: Find leading coefficient

Coefficient of highest degree term: $-4$

Answer:

$f(x)=-4x^4+\frac{5}{2}x^3+\sqrt{2}x^2+4x-6$
The degree is $4$.
The function is a quartic function.
The leading coefficient is $-4$.