Question

expressions-polynomials-evaluate add subtract
(1 point)
determine the following for: x08x08$x^2 - x^4 - 3x - 1$
a) determine the coefficient and the degree of each term.
\

$$\begin{tabular}{|c|c|c|} \\hline term & coefficient & degree \\\\ \\hline $x^2$ & \\boxed{} & \\boxed{} \\\\ \\hline $-x^4$ & \\boxed{} & \\boxed{} \\\\ \\hline $-3x$ & \\boxed{} & \\boxed{} \\\\ \\hline $-1$ & \\boxed{} & \\boxed{} \\\\ \\hline \\end{tabular}$$

b) the degree of the polynomial is \boxed{},
\quad the leading term is \boxed{},
\quad and the leading coefficient is \boxed{}.

Explanation:

Part (a)

For term \( x^2 \):

Step1: Find Coefficient

The coefficient is the numerical factor. For \( x^2 \), the coefficient is \( 1 \) (since \( x^2 = 1 \cdot x^2 \)).

Step2: Find Degree

The degree of a term with a single variable is the exponent of the variable. For \( x^2 \), the exponent is \( 2 \), so the degree is \( 2 \).

For term \( -x^4 \):

Step1: Find Coefficient

The coefficient is \( -1 \) (since \( -x^4 = -1 \cdot x^4 \)).

Step2: Find Degree

The exponent of \( x \) is \( 4 \), so the degree is \( 4 \).

For term \( -3x \):

Step1: Find Coefficient

The coefficient is \( -3 \) (since \( -3x = -3 \cdot x \)).

Step2: Find Degree

The exponent of \( x \) is \( 1 \) (since \( x = x^1 \)), so the degree is \( 1 \).

For term \( -1 \):

Step1: Find Coefficient

The coefficient is \( -1 \) (since \( -1 = -1 \cdot x^0 \), and \( x^0 = 1 \)).

Step2: Find Degree

The exponent of \( x \) (implicit) is \( 0 \), so the degree is \( 0 \).

Filling the table:

Term	Coefficient	Degree
\( -x^4 \)	\( -1 \)	\( 4 \)
\( -3x \)	\( -3 \)	\( 1 \)
\( -1 \)	\( -1 \)	\( 0 \)

Part (b)

Degree of the Polynomial:

The degree of a polynomial is the highest degree among its terms. From part (a), the highest degree is \( 4 \) (from the term \( -x^4 \)).

Leading Term:

The leading term is the term with the highest degree. The term with degree \( 4 \) is \( -x^4 \), so the leading term is \( -x^4 \).

Leading Coefficient:

The leading coefficient is the coefficient of the leading term. The leading term is \( -x^4 \), so its coefficient is \( -1 \).

Answer:

s:

Part (a) Table:

Term	Coefficient	Degree
\( -x^4 \)	\( -1 \)	\( 4 \)
\( -3x \)	\( -3 \)	\( 1 \)
\( -1 \)	\( -1 \)	\( 0 \)

Part (b)

• The degree of the polynomial is \( \boldsymbol{4} \).
• The leading term is \( \boldsymbol{-x^4} \).
• The leading coefficient is \( \boldsymbol{-1} \).