Question

exponents and polynomials
degree and leading coefficient of a univariate polynomial
what are the degree and leading coefficient of the polynomial?
18w⁷ − 10 + 8w² + w³
degree:
leading coefficient:

Explanation:

Step1: Rearrange the polynomial

First, we rearrange the polynomial in descending order of the exponents of \( w \). The given polynomial is \( 18w^{7}-10 + 8w^{2}+w^{3} \). Rearranging the terms by the degree of \( w \) (from highest to lowest), we get \( 18w^{7}+w^{3}+8w^{2}- 10 \).

Step2: Determine the degree

The degree of a polynomial is the highest power (exponent) of the variable in the polynomial. In the rearranged polynomial \( 18w^{7}+w^{3}+8w^{2}-10 \), the exponents of \( w \) are \( 7 \), \( 3 \), \( 2 \), and the constant term (which can be thought of as having an exponent of \( 0 \) for \( w \)). The highest exponent is \( 7 \), so the degree of the polynomial is \( 7 \).

Step3: Determine the leading coefficient

The leading coefficient is the coefficient of the term with the highest degree (the leading term). The leading term in the rearranged polynomial \( 18w^{7}+w^{3}+8w^{2}-10 \) is \( 18w^{7} \), and the coefficient of this term is \( 18 \). So the leading coefficient is \( 18 \).

Answer:

Degree: \( 7 \)
Leading coefficient: \( 18 \)