A factor is a place where the curve hits zero.

Here is $f(x) = x^3 - 2x^2 - 5x + 6$. Slide along it and watch the height: every time it drops to exactly zero, you've found a factor.

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Divide $f(x)$ by $(x - k)$ and the remainder is exactly $f(k)$. So wherever the curve crosses the axis, the division comes out clean.

$f(1) = 1 - 2 - 5 + 6 = 0$, so $(x - 1)$ is a factor.

$f(2) = 8 - 8 - 10 + 6 = -4$: dividing by $(x - 2)$ would leave remainder $-4$, so it isn't.

Dividing by $(x - 1)$ leaves $x^2 - x - 6$ with no remainder — and that quadratic factorises too:

Three factors, three roots: $-2$, $1$ and $3$, all clean crossings. $f(0) = 6$ gives the $y$-intercept, and the $x^3$ term sends it from bottom-left to top-right.

Find one root, and the rest follow.

The grid method, the remainder theorem, repeated roots, and twelve graded exam questions with full worked solutions are waiting.

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