Year 12 Algebra: Long division of polynomials
Back to Algebra tutorialsPolynomial long division is the algebraic version of ordinary long division. The goal is to rewrite a complicated polynomial fraction or expression in a more useful form by separating out the quotient and, if needed, the remainder. At first the layout can feel mechanical, but the logic is simple: each step removes the highest remaining power until there is nothing left to divide except a lower-degree remainder.
Subtopic 1: Standard long division
In standard long division, you compare the leading term of the dividend with the leading term of the divisor. That tells you the first term of the quotient. You then multiply back, subtract, and repeat. The process works because each subtraction removes the highest power still present. If you keep the columns aligned, the arithmetic becomes much safer.
Worked example 1
- Divide the leading terms: .
- Put in the quotient and multiply back: .
- Subtract from the dividend. This leaves .
- Now divide the new leading terms: .
- Put in the quotient and multiply back: .
- Subtract to get remainder .
Subtopic 2: Missing powers and careful setup
A very common Year 12 issue is forgetting a missing term. If a polynomial skips a power, you should still leave a placeholder when organising the division. This keeps the columns aligned and avoids subtracting the wrong terms later. Even when you do not literally write zero coefficients, you should be thinking in that structured way.
Worked example 2
- Notice the missing term, so think of the dividend as .
- First quotient term: .
- Multiply back: .
- Subtract to get .
- Next quotient term: .
- Multiply back: .
- Subtract to get .
- Final quotient term: .
- Multiply back: .
- Subtract to get remainder .
Why the remainder matters
The remainder is not just leftover clutter. It tells you that the divisor is not a factor of the original polynomial. When you later rewrite a rational expression, the remainder appears as a smaller fraction over the original divisor. This is useful in algebra and in some function contexts because it separates the expression into a polynomial part plus a simpler remainder term.
Long division also improves your sense of polynomial structure. You start to see which leading terms control the shape of the quotient, and you become much more comfortable rewriting expressions in ways that expose useful information. That is why the layout matters. A neat setup is not cosmetic here. It is part of the mathematics, because it keeps the powers lined up and the subtractions meaningful.
Common traps
- Skipping a missing power and misaligning the division.
- Subtracting without distributing the negative sign correctly.
- Stopping before the remainder has lower degree than the divisor.
- Forgetting to rewrite the final result as quotient plus remainder if required.
Revision focus
In revision, practise laying out the division slowly before trying to speed it up. Once the structure is stable, the arithmetic becomes much more reliable. It is also useful to check each answer by multiplying the divisor and quotient and then adding the remainder. If you recover the original polynomial, the division is correct.
Another good exercise is to explain aloud why each quotient term was chosen. If you can justify each step from the leading terms, you understand the process rather than just imitating the shape of the algorithm.