Skill: Determine the leading coefficient of a polynomial function

To determine the leading coefficient of a nonzero polynomial:

1. Write the polynomial in standard form, with exponents in descending order.
2. Find the term with the greatest exponent.
3. Identify the numerical factor multiplying that term. This number is the leading coefficient.

For example, consider

$$f(x)=4-7x^3+\frac{1}{2}x^5+2x^2.$$

First, rewrite the terms in descending order of exponents:

$$f(x)=\frac{1}{2}x^5-7x^3+2x^2+4.$$

The term with the greatest exponent is

$$\frac{1}{2}x^5.$$

Therefore:

• The leading term is $\frac{1}{2}x^5$.
• The degree is $5$.
• The leading coefficient is $\boxed{\frac{1}{2}}$.

The leading coefficient is the number attached to the term with the highest power of the variable. If no number is written, the coefficient is understood to be $1$; if the term begins with a negative sign, the coefficient is negative.