Question

Find the degree, leading coefficients, and the maximum number of real zeros of the polynomial.
F(X)= -3X^4-6X^5+2+4X^2


Degree =


Leading Coefficient =


Maximum number of real zeros =

Answer 1

To analyze the polynomial \( F(X) = -3X^4 - 6X^5 + 2 + 4X^2 \), we first rewrite it in standard form, arranging the terms in order of descending powers of \( X \):

\[
F(X) = -6X^5 - 3X^4 + 4X^2 + 2
\]

1. **Degree**: The degree of a polynomial is the highest power of \( X \) in the polynomial. In this case, the highest power is \( 5 \), so the degree is:

\[
\text{Degree} = 5
\]

2. **Leading Coefficient**: The leading coefficient is the coefficient of the term with the highest degree. In our polynomial, the highest degree term is \( -6X^5 \). Thus, the leading coefficient is:

\[
\text{Leading Coefficient} = -6
\]

3. **Maximum Number of Real Zeros**: The maximum number of real zeros of a polynomial is equal to its degree. Since the degree of this polynomial is \( 5 \), the maximum number of real zeros is:

\[
\text{Maximum number of real zeros} = 5
\]

To summarize:

- Degree = 5
- Leading Coefficient = -6
- Maximum number of real zeros = 5