Polynomial Division

1Summary 重點摘要

Just like primary long division, just numbers become x. I'll teach you step by step, no skipping.

2Vocabulary cards Vocabulary

3Step by step Steps

1. What division really asksPolynomial division answers the question: what must (x + 2) be multiplied by to give x² + 5x + 6? This is exactly like primary-school 126 ÷ 3 asking '3 times what makes 126?'. Here x² + 5x + 6 is the dividend, x + 2 the divisor, and the answer the quotient; anything left over is the remainder. With this in mind, every step of long division becomes a step closer to the answer rather than a mechanical ritual.
2. Step one: divide the leading termsLong division always starts at the top: divide the leading term of the dividend by the leading term of the divisor. For x² + 5x + 6 divided by x + 2, the leading terms are x² and x, and x² ÷ x = x becomes the first term of the quotient, written above the division sign. Never start from the lowest term or the alignment collapses; keeping the order from the highest power down is the soul of the long-division layout.
3. Step two: multiply the quotient term by the divisorMultiply the quotient term just found by every term of the divisor: x × x = x² and x × 2 = 2x, giving x² + 2x, written directly under the dividend with like powers aligned. This step computes the part already accounted for; the alignment matters because the next step subtracts like terms. If 2x drifts into the constant column, the subtraction will wrongly mix different powers and everything after it fails.
4. Step three: subtract and bring downSubtract the product from the dividend: x² + 5x minus x² + 2x cancels the x² terms and leaves 3x; then bring down the unused constant +6 to form 3x + 6. The classic error here is adding instead of subtracting to get 7x; the minus sign must apply to every term of the product. Once the next term is brought down, what remains becomes the target of the next round, and the cycle continues.
5. Step four: repeat until exactRepeat on 3x + 6: the leading 3x ÷ x = 3 joins the quotient; multiplying 3 by the divisor gives 3x + 6, and subtracting leaves (3x + 6) − (3x + 6) = 0. A zero remainder means the division is exact and the quotient is x + 3. The cycle is divide, multiply, subtract, bring down, continuing until what remains has degree below the divisor; any leftover is the remainder, and both quotient and remainder must be stated.
6. Verification and usesThe check formula is dividend = quotient × divisor + remainder. Here (x + 3)(x + 2) = x² + 5x + 6 matches the dividend, proving the answer. Polynomial division is widely used in algebra: it is a powerful tool for factorisation — knowing (x − 1) is a factor of some polynomial, division finds the other factor — and it becomes essential later with the remainder theorem and curve sketching, so practising it thoroughly pays off.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Polynomial division is like primary long division, numbers become x
• Always divide highest power first
• After multiplying, always subtract
• After subtracting, bring down next term and repeat
• Divide until subtract to 0