Question

exponents and polynomials
degree and leading coefficient of a univariate polynomial
what are the leading coefficient and degree of the polynomial?
$1 + 4y - 4y^5$

Explanation:

Step1: Identify the leading term

A polynomial's leading term is the term with the highest degree. For the polynomial \(1 + 4y - 4y^{5}\), we compare the exponents of \(y\) in each term. The exponents are \(0\) (for \(1 = 1y^{0}\)), \(1\) (for \(4y = 4y^{1}\)), and \(5\) (for \(-4y^{5}\)). The highest exponent is \(5\), so the leading term is \(-4y^{5}\).

Step2: Find the leading coefficient

The leading coefficient is the coefficient of the leading term. For the leading term \(-4y^{5}\), the coefficient is \(-4\).

Step3: Determine the degree

The degree of a polynomial is the highest power (exponent) of the variable in the polynomial. From the leading term \(-4y^{5}\), the exponent of \(y\) is \(5\), so the degree is \(5\).

Answer:

Leading coefficient: \(-4\)
Degree: \(5\)