7.2PureStretch

Dividing Polynomials

Polynomial long division works just like the long division you learned for numbers. It lets you break a complicated polynomial into a quotient and a remainder — the gateway to factorising cubics and beyond.

30 min Video by Zeeshan Zamurred Algebraic Methods

Edexcel AS Level Maths: 7.2 Dividing PolynomialsWatch the full walkthrough before the notes below.

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What you'll be able to do

Divide a polynomial by a linear expression
Identify the quotient and the remainder
Lay out polynomial long division clearly
Use division to help factorise higher polynomials

The long-division layout

Set out the division exactly like numerical long division. Divide the leading terms, multiply back, subtract, then bring down the next term — repeating until the remainder has a lower degree than the divisor.

dividend = divisor \times quotient + remainder

The identity every division satisfies — useful for checking your answer.

A worked division

Keep terms in descending powers and line up like terms. Take care with signs when subtracting each line.

x^{3} \div x = x^{2}

; multiply:

x^{2} (x - 2) = x^{3} - 2 x^{2}

; subtract to get

4 x^{2} - 5 x

4 x^{2} \div x = 4 x

; multiply:

4 x (x - 2) = 4 x^{2} - 8 x

; subtract to get

3 x - 6

3 x \div x = 3

; multiply:

3 (x - 2) = 3 x - 6

; subtract to get

0

Answerquotient

x^{2} + 4 x + 3

, remainder

0

Tip — A remainder of 0 means the divisor is a factor — exactly what the factor theorem (next lesson) exploits.

Missing terms

If a power is missing from the dividend, include it with a coefficient of $0$ as a placeholder (e.g. write $x^{3} + 0 x^{2} + \dots$ ). This keeps the columns aligned and prevents slips.

Formula recap

F (x) = D (x) Q (x) + R

Dividend = divisor × quotient + remainder.

de g (R) < de g (D)

Stop when the remainder’s degree drops.

Common mistakes to avoid

Sign errors when subtracting each line.

Subtracting flips the signs of the whole line — do it carefully, term by term.

Skipping a missing power and misaligning columns.

Insert 0x² (etc.) as a placeholder for any missing term.

Key takeaways

Polynomial long division mirrors numerical long division.
Divide leading terms, multiply back, subtract, bring down — repeat.
dividend = divisor × quotient + remainder; remainder 0 ⇒ a factor.
Use 0-coefficient placeholders for missing powers.

Test yourself

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