Here is a habit that will save you marks on every algebra paper you will ever write. Before you multiply two polynomials, pause for five seconds and predict the degree of the answer. When you finish expanding, check the degree of what you wrote. If it matches your prediction, move on. If it doesn't match, you lost a term somewhere — go back and find it.

That five-second habit is built on one clean rule: when you multiply polynomials, their degrees add. If P(x) has degree m and Q(x) has degree n, then P(x) · Q(x) has degree m + n. No exceptions that matter in practice. You can use this as a cheap, instant sanity check on every multiplication you do.

The rule — degrees add

Here is why deg(P · Q) = deg(P) + deg(Q).

Write the polynomials with their leading terms out in front:

When you multiply, the highest-degree term of the product comes from multiplying the two leading terms together:

(a·x^m) · (b·x^n) = a·b · x^(m+n)

Every other product — leading of P times a lower term of Q, or vice versa, or two lower terms — gives something of degree strictly less than m + n. So the x^(m+n) term stands alone at the top, with coefficient a·b.

The only way this could fail is if a·b = 0, which requires a = 0 or b = 0. But a polynomial whose leading coefficient is zero isn't actually that degree — you'd just rename it to be smaller. So for any two nonzero polynomials, the degrees add. Exactly. Always.

Worked example — catching a missed term

Multiply (x² + 3x)(x³ - 2).

Predicted degree: 2 + 3 = 5. Say that aloud: "answer should be degree 5."

Now expand:

Sum: x⁵ + 3x⁴ - 2x² - 6x.

Leading term is x⁵. Degree 5. Matches the prediction. You are safe.

Now suppose you had rushed and accidentally wrote 3x³ instead of 3x⁴, giving x⁵ + 3x³ - 2x² - 6x. Still degree 5 — that specific error wouldn't trip the check. But if you had dropped the x²·x³ term entirely and written only 3x⁴ - 2x² - 6x, your answer is degree 4. Check says: predicted 5, got 4. Error flagged. Go back.

The check doesn't catch every possible mistake, but it catches the most common one: a dropped term, especially the leading one.

Worked example — polynomial with apparent cancellation

Try (x² - x - 2)(x² + x + 2).

Predicted degree: 2 + 2 = 4.

Expand piece by piece so nothing is missed:

Now collect by degree:

Answer: x⁴ - x² - 4x - 4.

Notice the x³ terms cancelled. That is fine — it did not touch the leading x⁴. Degree 4. Matches prediction.

Middle terms can cancel. The leading term, for nonzero polynomials, cannot.

Quick drill — predict before expanding

Speak the predicted degree before doing any expansion.

Notice how quickly the prediction comes. You don't even need to know what the middle terms are. You just look at the highest power in each bracket and add.

Bonus — total term count

There is a second, weaker check you can do alongside the degree check.

If P has p terms and Q has q terms, the product has at most p · q terms before you combine like terms. After combining, some may merge. So if your pre-combination expansion has more than p · q terms, you have duplicated something. If it has fewer, you have skipped something.

For (x² + 3x)(x³ - 2): p = 2, q = 2, so up to 4 pre-combination terms. Count the pieces you wrote above — exactly 4. Good.

The degree bound is exact. The term-count bound is an upper limit. Together they form a very tight net for catching expansion errors.

When the prediction could be wrong

The only loophole: the product of the leading coefficients equals zero. For two polynomials of degrees m and n, this needs a · b = 0, which needs a = 0 or b = 0. But if a = 0, your polynomial wasn't really degree m — it was lower. So for any two nonzero polynomials written correctly, deg(P · Q) = deg(P) + deg(Q). No exceptions you will encounter.

You may sometimes see identities like (x - 1)(x² + x + 1) = x³ - 1 where lower-degree terms cancel. That is fine — the leading term still stands.

Use it as your confidence check

Every polynomial multiplication in your workings should end with one silent sentence: "degree should be m + n; I got m + n; move on."

If the degrees disagree, don't try to patch the last line. Go back to the expansion, write out every pairwise product in a grid, and find the missing one. The error is almost always a dropped leading-times-leading or a sign flip in the highest term.

Degree addition is more than a check — it's a structural property

Once you see that degrees add under multiplication, some larger facts fall out for free.

That last one is useful in JEE problems where you need to know the highest power of x in a huge expansion without actually expanding it.

Even for multivariable polynomials

The rule extends cleanly. The "degree" of a multivariable polynomial is the highest total degree among its terms — the sum of all variable exponents in a single term.

Same rule, same check.

Recognition drill

State the predicted degree before expanding. No computation needed beyond adding the top exponents.

You should be able to say these numbers faster than you can read the expression.

The habit

Predict the degree before you expand. Say it aloud. When you finish expanding, check the leading term matches. If it does, your answer is almost certainly correct. If it doesn't, you lost a term — and you now know exactly where to look.

Five seconds of prediction saves five minutes of re-doing the problem. Make it automatic.