Question

homework assignment 3.3: power functions and polynomial functions
score: 1/10 answered: 1/10
question 2
find the degree, leading coefficients, and the maximum number of real zeros of the polynomial.
$f(x) = -2x^4 + 5x^2 - 3 - x^5$
degree =
leading coefficient =
maximum number of real zeros =
question help: video written example

Explanation:

Step1: Rearrange the polynomial

First, we rearrange the polynomial \( f(x) = -2x^4 + 5x^2 - 3 - x^5 \) in descending order of exponents: \( f(x)= -x^5 - 2x^4 + 5x^2 - 3 \).

Step2: Find the degree

The degree of a polynomial is the highest power of \( x \) with a non - zero coefficient. In the polynomial \( f(x)= -x^5 - 2x^4 + 5x^2 - 3 \), the highest power of \( x \) is \( 5 \), so the degree is \( 5 \).

Step3: Find the leading coefficient

The leading coefficient is the coefficient of the term with the highest degree. For the term \( -x^5 \), the coefficient is \( - 1 \), so the leading coefficient is \( -1 \).

Step4: Find the maximum number of real zeros

By the Fundamental Theorem of Algebra, the maximum number of real zeros of a polynomial of degree \( n \) is \( n \). Since the degree of the polynomial is \( 5 \), the maximum number of real zeros is \( 5 \).

Answer:

Degree = \( 5 \)
Leading Coefficient = \( - 1 \)
Maximum number of real zeros = \( 5 \)