Degree and Leading Coefficient of a Polynomial

Before you can divide a polynomial, find its zeros, or sketch it, you need to know two numbers about it: its degree and its leading coefficient. Together they fix the end behavior of the graph, cap the number of zeros and turning points it can have, and tell you what kind of polynomial you are dealing with. This lesson is about reading those two numbers off any polynomial reliably — including the ones written in a scrambled order, in factored form, or with more than one variable, where the obvious answer is often wrong.

What Is the Degree of a Polynomial?

A polynomial in one variable is a sum of terms, each of which is a number times a whole-number power of :

The degree of a polynomial is the highest power of the variable that appears with a non-zero coefficient. In the general form above, that is . The degree of each individual term is its exponent; the degree of the polynomial is the largest of those.

The last row is worth pausing on. A constant is a polynomial too — — and its degree is . The only exception is the zero polynomial, , which has no non-zero term to take the degree from, so its degree is undefined.

The Leading Term, Leading Coefficient, and Constant Term

Once the polynomial is written in descending powers, three parts of it get names:

• The leading term is the term with the highest degree: .
• The leading coefficient is the number in front of that term: , sign included.
• The constant term is the term with no variable: . If none is written, the constant term is .

For :

$$ \underbrace{4x^{5}}{\text{leading term}} - 3x^{2} + \underbrace{7}{\text{constant term}}, \qquad \text{degree } 5, \qquad \text{leading coefficient } 4 $$

The leading coefficient of a polynomial is a single number, and its sign matters. In the leading coefficient is , not and not “there isn’t one”. An unwritten coefficient is always or , depending on the sign in front.

How to Find the Degree of a Polynomial

Three situations cover almost every polynomial you will meet.

Case 1 — already in descending order. The degree is the exponent of the first term and the leading coefficient is its coefficient. For : degree , leading coefficient .

Case 2 — written out of order. Scan every term for the largest exponent, or better, rewrite the polynomial in standard form first. For , the largest exponent is , so:

Reading the first term you see, , and calling it the leading term is the single most common mistake with this topic.

Case 3 — factored form. You do not need to expand the whole product. The degree is the sum of the degrees of the factors, and the leading coefficient is the product of their leading coefficients. For

the factor degrees are , and the leading coefficients multiply to . So has degree and leading coefficient , and its expanded form would begin .

A fourth case comes up less often: more than one variable. Then the degree of each term is the sum of its exponents, and the degree of the polynomial is the largest of those sums. In , the term degrees are , so the polynomial has degree . “Leading coefficient” is only well-defined for one-variable polynomials, so it is rarely asked about here.

Classifying Polynomials by Degree and Number of Terms

Two classification systems run side by side, and a polynomial gets a name from each.

By degree:

By number of terms (after combining like terms):

So is a quadratic binomial, and is a cubic binomial. The two labels are independent — a cubic can be a monomial, a binomial, or a trinomial.

Is It a Polynomial at All?

The degree only makes sense if the expression is actually a polynomial, and some expressions that look like one are not. Every exponent on the variable has to be a whole number (), and the variable can never sit in a denominator or under a root.

The rule is about the exponents on the variable, not about the coefficients. Coefficients can be any real numbers at all.

Worked Example: A Polynomial Written Out of Order

Find the degree, leading coefficient, leading term, and constant term of , and classify it.

Step 1 — identify the exponent on each term:

Step 2 — rewrite in descending order of exponent:

Step 3 — read the parts off the front and the back:

• Leading term:
• Degree:
• Leading coefficient:
• Constant term:

Step 4 — classify. Degree makes it quartic; four terms means it is simply a four-term polynomial (there is no special name past three).

Note that the term is missing. That does not affect anything here — a missing power just has coefficient — though it matters later when you set up polynomial long division, where the gap has to be held open with a placeholder.

Worked Example: Degree and Leading Coefficient From Factored Form

Find the degree and leading coefficient of without expanding.

Step 1 — take the degree of each factor and add:

$$ \underbrace{x^{2}}{2} \quad \underbrace{(x - 2)^{3}}{3} \quad \underbrace{(4 - x)}_{1} \qquad\Longrightarrow\qquad 2 + 3 + 1 = 6 $$

Step 2 — take the leading coefficient of each factor and multiply. This is where the trap is: the last factor is , whose leading term is , not .

Step 3 — state the answer. Degree , leading coefficient . Expanded, would begin .

Check. The highest-power term of the expansion comes only from multiplying the highest-power piece of every factor: . Every other product of pieces has a lower power, so the leading term really is .

What the Degree and Leading Coefficient Tell You

These two numbers are not just labels; each controls something visible on the graph.

End behavior. Whether the degree is even or odd, together with the sign of the leading coefficient, fixes what the two tails of the graph do. This is often called the leading coefficient test, and the end behavior lesson works through all four cases.

Maximum number of zeros. A polynomial of degree has at most real zeros — at most places where the graph crosses or touches the -axis. A cubic can cross three times, but never four.

Maximum number of turning points. The graph of a degree- polynomial has at most turning points. A quadratic turns once; a quartic turns at most three times.

The constant term is the -intercept. Setting kills every term except , so the graph crosses the -axis at .

All three counts are ceilings, not exact values — has degree but only one zero and one turning point. They tell you what a graph cannot do, which is exactly what you need when checking a sketch.

Common Mistakes to Avoid

• Taking the first term as the leading term. In the leading term is . Always look for the largest exponent, not the leftmost term.
• Dropping the sign of the leading coefficient. For the leading coefficient is . The sign is the half of the leading coefficient test that decides which way the right-hand tail points.
• Adding exponents within a single-variable term. The degree of is , not or anything else. Exponents only add across different variables in the same term, like , or across factors in a product.
• Confusing the degree of a term with its coefficient. In the degree is and the coefficient is . “Degree” always means the exponent.
• Counting terms before simplifying. looks like a four-term quadratic but simplifies to , a linear binomial. Combine like terms before classifying.
• Ignoring a backwards factor. contributes a leading coefficient of . Rewriting it as before you multiply makes the sign impossible to lose.

Practice Problems

Work each one before opening the answer.

Problem 1. What is the degree and leading coefficient of ?

Problem 2. Find the degree, leading coefficient, and constant term of .

Problem 3. Classify by degree and by number of terms.

Problem 4. Without expanding, give the degree and leading coefficient of .

Problem 5. Which of these are polynomials? For each one that is, state its degree. (a) (b) (c) (d)

Problem 6. The graph of a polynomial has five turning points and both tails pointing down. What is the smallest possible degree, and what is the sign of the leading coefficient?

The degree and leading coefficient are the first two things to extract from any polynomial, and the habit of writing it in standard form before you do anything else is what keeps them from being misread. From here, the end behavior lesson puts the two numbers to work on the tails of the graph, and graphing polynomials uses them to sanity-check a full sketch.