Chapter 3 · Expansion & Factorisation

1.1.6 Dividing Polynomials

Dividing a polynomial by (x + 2) works exactly like dividing 851 by 23 — divide, multiply, subtract, bring down. Just with letters instead of digits.

🎯 By the end of this you can…

✓Lay out a polynomial division the same way as a number long division
✓Divide a cubic by a linear binomial, term by term
✓Check an answer by multiplying the quotient back by the divisor

💡 The idea

Same algorithm, different symbols

Long-dividing $x^{3} + x^{2} - 3 x - 2$ by $x + 2$ uses the exact three moves as dividing numbers: divide the leading terms, multiply that answer by the whole divisor, subtract. Then bring down the next term and repeat.

📋 The method

1
Write it as a long division, dividend inside, divisor outside.
2
Divide the FIRST term of what's left by the FIRST term of the divisor. Write that in the answer.
3
Multiply that answer by the whole divisor, then subtract the result from what's left (watch the signs!).

➗ ( $x^{3} + x^{2} - 3 x - 2$ ) ÷ ( $x + 2$ ) — one row at a time

⭐ 5 XP

1Divide:

x^{3}

x

x^{2}

Multiply:

x^{2}

(

x + 2

) =

x^{3} + 2 x^{2}

Subtract: (

x^{3} + x^{2} - 3 x - 2

) − (

x^{3} + 2 x^{2}

) =

- x^{2} - 3 x - 2

💡Why does subtracting always cancel the leading term? The quotient term was built BY dividing the leading term exactly — multiplying it back by the divisor's leading term always reproduces that same leading term, so it's guaranteed to cancel to 0 every time.

row 1 / 3

✏️ Worked example

(a³ + b³) ÷ (a + b)

1
$a^{3} \div a = a^{2}$ ; multiply $a^{2} (a + b) = a^{3} + a^{2} b$ ; subtract: $- a^{2} b + b^{3}$
2
$- a^{2} b \div a = - ab$ ; multiply $- ab (a + b) = - a^{2} b - a b^{2}$ ; subtract: $a b^{2} + b^{3}$
3
$a b^{2} \div a = b^{2}$ ; multiply $b^{2} (a + b) = a b^{2} + b^{3}$ ; subtract: $0$

✅ Answer: $a^{2} - ab + b^{2}, remainder 0$

⚠️ The classic trap

Subtracting a negative is where most mistakes happen. $(- x^{2} - 3 x) - (- x^{2} - 2 x)$ means flipping every sign in the second bracket before combining: $- x^{2} - 3 x + x^{2} + 2 x = - x$ . Writing it out sign by sign, every time, avoids the slip.

🎮 Find each quotient (every one divides exactly, remainder 0)

⭐ 0 XP🔥 streak 0

(

a^{2} + 2 ab + b^{2}

) ÷ (

a + b

)

(

8 x^{3} - 10 x^{2} - x + 3

) ÷ (

x - 1

)

(

x^{2} - 4

) ÷ (

x - 2

)

(

x^{2} + 5 x + 6

) ÷ (

x + 2

)

0 of 4 finished

🤔 How do you check that (x² − x − 1) really is the quotient of (x³ + x² − 3x − 2) ÷ (x + 2)?

🏅

You finished this lesson!

Polynomial long division is number long division wearing letters — the same three moves, repeated until nothing's left to bring down.

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