Polynomial Long Division

Polynomial long division is the bridge between finding one factor of a cubic and factorising it completely. Once the factor theorem gives you a linear factor, dividing by it leaves a quadratic quotient you can factorise the usual way.

The routine

Divide $x^3 - 3x^2 - 4x + 12$ by $(x - 2)$.

Work term by term, highest power first:

1. $x^3 \div x = x^2$. Multiply back: $x^2(x - 2) = x^3 - 2x^2$. Subtract: $(x^3 - 3x^2) - (x^3 - 2x^2) = -x^2$.
2. Bring down: $-x^2 - 4x$. Now $-x^2 \div x = -x$. Multiply back: $-x(x - 2) = -x^2 + 2x$. Subtract: $(-x^2 - 4x) - (-x^2 + 2x) = -6x$.
3. Bring down: $-6x + 12$. Now $-6x \div x = -6$. Multiply back: $-6(x - 2) = -6x + 12$. Subtract: $0$.

The quotient is $x^2 - x - 6$, with remainder $0$ (confirming $(x - 2)$ is a factor): $x^3 - 3x^2 - 4x + 12 = (x - 2)(x^2 - x - 6)$

Comparing coefficients: the faster alternative

Many students prefer to skip long division and compare coefficients. Write $(x - 2)(x^2 + bx + c)$, expand, and match: $(x - 2)(x^2 + bx + c) = x^3 + (b - 2)x^2 + (c - 2b)x - 2c$

Matching $x^3 - 3x^2 - 4x + 12$: $-2c = 12 \Rightarrow c = -6$; $b - 2 = -3 \Rightarrow b = -1$. So the quadratic is $x^2 - x - 6$, the same result, often more quickly.

Common mistakes

• Sign errors when subtracting each partial product (subtracting a negative).
• Losing a term by not lining up powers (use a $0x$ placeholder if a power is missing).
• Forgetting to factorise the quadratic quotient afterward when “completely” is asked.

Full topic context: Factors of Polynomials notes, and the worked cubic example.