Question

exponents and polynomials
degree and leading coefficient of a univariate polynomial
what are the degree and leading coefficient of the polynomial?
$-y^5 - 9y^2 + 2 - 5y$
degree:
leading coefficient:

Explanation:

Step1: Find the degree of the polynomial

The degree of a polynomial is the highest power (exponent) of the variable in the polynomial. In the polynomial \(-y^{5}-9y^{2}+2 - 5y\), we look at the exponents of \(y\) in each term:

• For the term \(-y^{5}\), the exponent of \(y\) is \(5\).
• For the term \(-9y^{2}\), the exponent of \(y\) is \(2\).
• For the term \(2\) (which can be written as \(2y^{0}\)), the exponent of \(y\) is \(0\).
• For the term \(-5y\) (which can be written as \(-5y^{1}\)), the exponent of \(y\) is \(1\).

The highest exponent among these is \(5\), so the degree of the polynomial is \(5\).

Step2: Find the leading coefficient of the polynomial

The leading coefficient is the coefficient of the term with the highest degree (the leading term). The leading term in the polynomial \(-y^{5}-9y^{2}+2 - 5y\) is \(-y^{5}\) (since it has the highest degree, \(5\)). The coefficient of \(y^{5}\) in this term is \(- 1\) (because \(-y^{5}=-1\times y^{5}\)). So the leading coefficient is \(-1\).

Answer:

Degree: \(5\)
Leading coefficient: \(-1\)