Why does FOIL stop working?

Because it names four products, and only two binomials have exactly four. A binomial times a trinomial has six, and the acronym has no letter for the fifth and sixth. The underlying rule — every term times every term — never stops working, and the grid method is that rule written down without a case limit.

How many terms should my answer have after multiplying?

At most m × n, where m and n are the term counts of the two factors, and usually fewer because like terms collect. Counting the products before you start is the check: if you wrote eight where there should have been nine, one is missing, and nothing later in the working will reveal it.

Do I have to write the zeros for missing powers?

For synthetic division, yes — the columns carry the degree and a gap shifts everything after it. For long division it is strongly advisable for the same reason, since the method is entirely about keeping like terms aligned. For adding and subtracting it is optional in a line and worth doing in columns.

What is the remainder theorem for?

It gives you the remainder of a division by (x − c) without doing the division: just evaluate the polynomial at c. That is far quicker when the remainder is all you want, and it is the basis of the factor theorem — a remainder of zero means (x − c) divides exactly, so c is a root.

Polynomial Arithmetic: Add, Multiply, Divide

Q: How do I check a polynomial answer quickly?

Substitute a number. Put x = 2 into the original expression and into your answer; an identity holds for every value of x, so they must agree. It is not a proof, but a disagreement is a definite error and the check takes ten seconds. For a division, multiply the quotient by the divisor and add the remainder — that must return the dividend exactly.

Adding and subtracting: the minus belongs to the bracket

Adding polynomials is gathering like terms and nothing else — terms are like when their variable parts match exactly, so 3x²y and 5x²y combine while 3x²y and 3xy² do not, however similar they look.

Subtracting is the same job with one step in front of it, and that step is where the marks go. (3x² + 2x − 5) − (x² − 4x + 1) is not 3x² + 2x − 5 − x² − 4x + 1. The minus applies to the whole bracket, so every sign inside flips: −x² + 4x − 1. The answer is 2x² + 6x − 6. Distributing the minus over only the first term gives 2x² − 2x − 4, which is wrong and looks entirely reasonable.

Worked example

(3x² + 2x − 5) − (x² − 4x + 1) = 3x² + 2x − 5 − x² + 4x − 1 = 2x² + 6x − 6

Why the vertical layout is worth the extra writing

Written in a line, a missing term is invisible. 5x³ − 2x + 7 has no x² term, and when you subtract 3x³ + x² − 2x from it in a single line there is nothing on the page to remind you that the x² column has an entry on only one side.

Stacking the polynomials with like terms in columns turns that into a visible gap. It costs a few seconds and it removes an entire class of error — the same reason long multiplication with digits is written in columns rather than in a line.

Multiplying: count the products before you start

Multiplication is one rule — every term in the first bracket times every term in the second — and one number worth writing down first: m terms times n terms is m×n products, always, before anything is collected. Two binomials give four. A binomial times a trinomial gives six. Two trinomials give nine.

Knowing that number in advance is the only reliable way to notice that you wrote eight when there should have been nine, and it is why the grid method survives past the point where FOIL stops. A grid with one empty cell is obviously incomplete; a line with one missing product is not.

m terms × n terms = m × n products, before collecting

FOIL is a special case, not a method

FOIL names the four products of two binomials — First, Outer, Inner, Last — and it works because two-times-two is four. It does not extend. (x + 2)(x² + 3x + 4) has six products, and there is no letter for the fifth and sixth, which is why students who learned only FOIL stall at exactly this problem.

The mnemonic is fine for what it covers. The trap is treating it as the method rather than as a name for one case of the distributive law. The grid is the same distribution with no case limit, and it is worth switching to it before you need to rather than after.

Worked example

(x + 2)(x² + 3x + 4): six products — x³, 3x², 4x, 2x², 6x, 8 — collecting to x³ + 5x² + 10x + 8

The degree check, which takes five seconds

When you multiply polynomials, the degrees add. A cubic times a quadratic is a quintic, always, because the leading terms multiply and nothing else can reach that degree. If your answer's degree is not the sum of the factors' degrees, a leading term was dropped or mis-multiplied.

It is the cheapest check in polynomial arithmetic and it catches the errors that matter most, because an error in the leading term changes the shape of everything downstream — a factorisation, a root, a limit.

Dividing: long division always, synthetic sometimes

Long division works for any divisor and runs the same way as it does with numbers: divide the leading term by the leading term, multiply back, subtract, repeat. The subtraction is the failure point — you subtract the whole product, so every one of its signs flips, and forgetting that from the second term onward is the classic error.

Synthetic division is a shorthand that works only when the divisor is linear. It drops the variables entirely and works with the coefficients, which is what makes it fast and also what makes a missing power fatal: the columns carry the degree, so an absent x² term must be written in as a zero or everything after it shifts.

Two theorems come free. Dividing by (x − c) leaves P(c) as the remainder, so you can find a remainder without dividing at all. And that remainder is zero exactly when (x − c) is a factor — which is how a root and a factor are the same fact wearing different clothes.

P(x) = D(x)·Q(x) + R(x), with deg R < deg D

One check for all four operations

Every one of these operations can be checked by reversing it. A product can be checked by re-expanding, a division by multiplying the quotient back by the divisor and adding the remainder, a subtraction by adding the answer back to what was subtracted.

A faster check when you only need confidence rather than proof: substitute a number. Put x = 2 into the original expression and into your answer. They must agree, because an identity holds for every value of x. Two different numbers agreeing is not a proof, but a disagreement is a definite error, and it takes about ten seconds.

Common mistakes

Distributing a subtraction over only the first term of the bracket. The minus belongs to the whole bracket and reaches the last term as surely as the first.
Treating x²y and xy² as like terms. Like terms need identical variable parts, exponents included.
Using FOIL where a bracket has three terms. Two-times-three is six products and the acronym names four of them.
Writing a polynomial for division without filling missing powers with zeros. In synthetic division the columns carry the degree, so a gap shifts everything after it.
Forgetting to flip every sign of the product when subtracting in long division. This is the same bracket error as in ordinary subtraction, one layer down.
Stopping a division while the remainder still has degree at least that of the divisor. It always goes one stage further than it feels like it should.

When not to rely only on the calculator

This is arithmetic on polynomials with exact coefficients. Where the coefficients come from measurements or fitted data, the exact answer may be less useful than it looks: dividing polynomials with approximate coefficients amplifies error quickly, and a factorisation that depends on a coefficient being exactly 3 rather than 3.0001 is not a fact about the data. Nothing here covers rational expressions with variables in the denominator either — cancelling those has its own rule about excluded values, and dropping it changes the domain of the expression.

Adding, Multiplying and Dividing Polynomials — the Four Operations