Dividing polynomials by (x − k), the long way

A typical question:

Divide $2x^3 - 5x^2 + 3x - 4$ by $x - 2$, stating the quotient and the remainder.

The answer always has the same shape:

$\text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder}$

Two facts make the arithmetic predictable:

• when the divisor is linear, the remainder is a constant (possibly zero);
• the quotient is one degree lower than the dividend, so a cubic gives a quadratic quotient.

The same words appear in ordinary arithmetic. Dividing $17$ by $5$ gives quotient $3$ and remainder $2$, because $17 = 5 \times 3 + 2$.

1. Set out the division with the dividend inside and the divisor outside, both in descending powers.
2. Write $0x$ for any missing power, so that every column exists.
3. Divide the leading terms and write the result on the top row — that is the next term of the quotient.
4. Multiply the whole divisor by that new term.
5. Subtract the product from what is currently inside.
6. Bring down the next term, then repeat with what is left, until the remaining degree is less than the divisor's.
7. Name the parts: the top row is the quotient, and the bottom line is the remainder.

The forgotten step is step 2. Leaving out a missing power misaligns every column after it, and the arithmetic then goes wrong in a way that is hard to spot.

Subtract, do not add. The signs reverse on every term of the product, and a single slip there is the most common error in the whole topic.