Here is a secret hiding in plain sight: you already know the algorithm this page teaches. You learned it when you were about eight, chanting divide, multiply, subtract, bring down over sums like 736 \div 3 — that's long division. Polynomial long division is that exact same dance, wearing algebra's clothes. Where you once asked "how many times does 3 go into 7?", you now ask "how many times does x go into x^3?" — and every other move is identical. If you can divide 736 by 3, you can divide x^3 + 7x^2 + 3x + 6 by x + 3.

Why bother? Because division is the reverse of multiplying polynomials, and reversing a multiplication is how you take polynomials apart. Long division is the workhorse behind factorising cubics, testing whether (x - a) is a factor, and rewriting awkward algebraic fractions into friendlier pieces — skills you will lean on constantly in calculus and beyond.

The vocabulary carries straight over from arithmetic, too. The polynomial being divided is the dividend, the polynomial you divide by is the divisor, the answer is the quotient, and whatever is left over at the end — too small for the divisor to go into again — is the remainder. In 736 \div 3, "too small" means less than 3. In polynomial division it means lower degree than the divisor: dividing by the degree-1 divisor x + 3, you stop when only a constant is left.

The same dance, side by side

Don't take our word for it — watch the two divisions below move in lockstep. On the left, 736 \div 3 exactly as you set it out in primary school. On the right, (x^3 + 7x^2 + 3x + 6) \div (x + 3). Step through them together: every single move on the left has a twin on the right. The only difference is that the columns hold powers of x instead of powers of ten — which is no coincidence, since 736 literally is 7t^2 + 3t + 6 with t = 10.

Both divisions ended with something left over. On the left, 3 goes into 736 two-hundred and forty-five times with 1 spare:

And on the right, x + 3 goes into the cubic "x^2 + 4x - 9 times" with 33 spare:

The other standard way to write a division with a remainder is as quotient + remainder over divisor — just as 736 \div 3 = 245\tfrac{1}{3}, we write:

Learn both forms — exam questions ask for each. And notice the built-in check: expand (x + 3)(x^2 + 4x - 9) + 33 and you must land back on the dividend. Multiplying back is to division what addition is to subtraction — the free way to verify your answer.

The smallest complete example

Now step through the tidiest case there is: x^2 + 5x + 6 divided by x + 2. Two cycles of divide, multiply, subtract, bring down — and this time the subtraction at the end leaves exactly 0.

Two things to notice as you step. First, at each divide stage you only ever look at the leading terms — x^2 \div x, then 3x \div x — everything else waits its turn. Second, each subtract is engineered to kill the leading term, so what's left always has strictly smaller degree than before. That's why the algorithm can never run forever: the degree drops with every cycle until it falls below the divisor's, and then you're done.

A remainder of 0 is big news: it means x + 2 divides x^2 + 5x + 6 exactly, so it is a factor. Indeed factorising gives x^2 + 5x + 6 = (x + 2)(x + 3) — the quotient x + 3 is the other factor, delivered to you by the division.

Worked example: cracking a cubic open

This is the move you'll use most at A-level: someone hands you a cubic and one suspected factor, and long division breaks the cubic apart. Divide x^3 - 2x^2 - 5x + 6 by x - 1.

Cycle 1. Divide leading terms: x^3 \div x = x^2. Multiply back: x^2(x - 1) = x^3 - x^2. Subtract and bring down the -5x:

Cycle 2. Divide: -x^2 \div x = -x. Multiply back: -x(x - 1) = -x^2 + x. Subtract carefully — the second bracket's +x comes off — and bring down the +6:

Cycle 3. Divide: -6x \div x = -6. Multiply back: -6(x - 1) = -6x + 6. Subtract:

Remainder 0. So x - 1 is a factor, and the quotient is x^2 - x - 6:

And here is the payoff — the quotient is a mere quadratic, which you can factorise on sight: x^2 - x - 6 = (x - 3)(x + 2). The cubic falls completely apart:

One division turned an impenetrable cubic into three linear factors — and told you its roots are 1, 3 and -2. That is why polynomial division earns its keep.

The missing-term trap: x^3 + 1

Try to divide x^3 + 1 by x + 1 and the layout itself fights you. Write the dividend as just "x^3 + 1", and after the first cycle — multiply back x^2(x+1) = x^3 + x^2, subtract to get -x^2 — there is nothing to bring down into the x^2 and x columns, because those columns don't exist on your page. Most people then subtract the x^2 from the 1, as if x^2-stuff and constant-stuff lived in the same column. Crash.

It's exactly like writing 1006 as "16" because the middle digits are zero — the columns lie. The fix: write every missing power with a 0 coefficient placeholder,

and now the machine runs perfectly smoothly:

• x^3 \div x = x^2; subtract x^3 + x^2, bring down: -x^2 + 0x
• -x^2 \div x = -x; subtract -x^2 - x, bring down: x + 1
• x \div x = 1; subtract x + 1: remainder 0

You have just derived the famous sum of cubes factorisation — long division found it for you, no memorising required.

See it explained

For a second voice on the whole procedure, watch Sal Khan work through dividing a polynomial by a linear expression — same four moves, same layout, one more full example for your collection.