Finding the degree of a polynomial is a one-second job if you let it be one. You don't need to expand, you don't need to simplify, you don't need to think about what the polynomial "is doing". You scan the thing with your eyes, pick out the biggest exponent on a variable, and you are done. Everything else on the page — the constant at the end, the leading coefficient, the middle terms, the signs — is a distraction for this particular question.
Train this as a reflex. Degree questions should take less time than reading them.
The rule
The degree of a polynomial in one variable is the highest power of the variable that appears with a nonzero coefficient. For a polynomial in several variables, the degree is the highest total exponent (sum of exponents) across all the terms.
That's it. No sub-clauses, no edge cases worth memorising beyond the "nonzero coefficient" footnote. Every other piece of information in the polynomial is irrelevant to this one question.
Worked examples
Run your eye across each of these and ignore everything except the exponents.
3x² + 5x + 2— the exponents you see are 2, 1, 0. The biggest is 2. Degree is 2. You don't care about the 3, you don't care about the 5, you don't care about the 2.7x⁵ - x + 100— exponents 5, 1, 0. Biggest is 5. Degree 5.x + 4x⁷ - 2x²— the polynomial is written in non-standard order (the biggest term isn't first), but that doesn't matter. Exponents 1, 7, 2. Biggest is 7. Degree 7. Don't re-order the polynomial in your head — just find the biggest number floating up in the exponent slot.8— no variable visible. A constant is interpreted as8x⁰, and the degree is 0.0— the zero polynomial is the single edge case. Its degree is left undefined (some books define it as −∞ for algebraic convenience, but for your purposes: undefined).
Multi-variable case
When more than one variable shows up, add the exponents on each term, then pick the biggest sum.
For 3x²y + 2xy² + 5:
- First term: exponents 2 and 1, total 3.
- Second term: exponents 1 and 2, total 3.
- Third term: no variables, total 0.
Biggest total is 3. Degree of the polynomial is 3.
For 7x⁴y² + 2x³y⁵: totals are 6 and 8. Degree is 8.
Notice: you don't care which variable supplied the exponent, and you don't care how the exponents are split across the variables. Just add them up per term.
What NOT to waste time on
When someone asks you for the degree, here is a list of things that are not your problem:
- The constant term. Irrelevant.
- The leading coefficient (the number multiplying the highest-power term). Irrelevant to the degree, even though it looks important.
- Factoring the polynomial. Don't try to factor to find the degree — factoring is harder, and gives you the same answer.
- Simplifying. Don't expand products or combine unlike terms just to find the degree. Only exception: if two terms with the same exponent are both present and their coefficients might cancel.
If you catch yourself doing arithmetic to find the degree, stop. You've overshot.
The one subtlety — canceling terms
There is exactly one trap. Consider:
3x² - 3x² + 5x + 1
If you scan lazily, you see an x² and say "degree 2". But the two x² terms cancel — their coefficients add to zero. What's left is 5x + 1. Degree is 1, not 2.
So the careful statement of the rule is: degree = highest exponent with a nonzero coefficient after you've combined like terms. In real problems, polynomials are almost always given in already-simplified form and this trap doesn't fire. But keep a half-eye open for it. If you see two terms with the same exponent, glance at their coefficients.
The nonzero-coefficient check in practice
When a polynomial is given as a product of factors, you can sometimes see the degree without expanding — but check that the leading term doesn't cancel.
(x − 1)²— expanded this isx² − 2x + 1. Thex²term has coefficient 1, which is nonzero. Degree 2. (The sum of the factor degrees is 1 + 1 = 2 — same answer, faster.)(x − 1)(x + 1)(x − 2)— three linear factors. Leading term isx · x · x = x³, coefficient 1. Degree 3.(x² + 1)(x − 1)— leading term isx² · x = x³, coefficient 1. Degree 3.
In each case the leading coefficient is obviously nonzero, so the "degree = sum of factor degrees" shortcut just works.
Shortcuts
When the polynomial is written as a product of factors, the degree is the sum of the factor degrees. You don't have to expand.
(x + 1)²(x − 3)³ has degree 2 + 3 = 5.
(x² + 2x + 1)(x − 4)(x + 7) has degree 2 + 1 + 1 = 4.
(x³ − 1)² has degree 3 × 2 = 6 (a power of a polynomial multiplies its degree).
This shortcut is the one you will use most on JEE problems, where polynomials are often shipped as products. Don't expand. Add the degrees.
Recognition drill
Call out the degree for each of these in under a second:
5x³ − 2x + 7→ 3.17→ 0.(x − 1)(x − 2)(x − 3)(x − 4)→ 4 (four linear factors, sum 1+1+1+1).(x² − 5)(x² + 5)→ 4. The product isx⁴ − 25; thex⁴term does not cancel. Sum of factor degrees: 2 + 2.x² − x² + x + 1→ 1. Thex²terms cancel; what remains has biggest exponent 1.2x⁷y³ − xy⁹ + 5→ 10. Totals are 10, 10, 0.
If any of those took more than a heartbeat, read through them again and watch where your eye landed first. You want your eye to fly straight to the exponents.
Why we emphasise the degree
The degree of a polynomial is a ridiculously rich number for such a cheap computation. It tells you the end behaviour (how the graph shoots off at infinity), it bounds the number of real roots, it bounds the number of turning points in the graph, it determines how many coefficients the polynomial has, and it is the first thing a question will ask you for. You will read the degree more often than almost any other property of a polynomial in the rest of your algebra career. Drilling it into a one-scan reflex pays back every single time.
Don't overthink
Final shortcut. If the polynomial is already written in standard form — descending powers, like a JEE textbook writes them — the exponent of the first term is the degree. Full stop. Don't scan further.
If it is not in standard form, one sweep across the page finds the biggest exponent. Still seconds.
Closing
The degree is one property of the polynomial, and the question "what is the degree?" is answered by one action: find the biggest exponent. Don't compute the constant term. Don't check the leading coefficient. Don't try to factor. Don't expand. You were asked for one number — give one number, and move on to the harder part of the problem.