Highest Common Factor

The most basic form of algebraic factorisation is highest common factor, which involves taking out the largest possible term that divides evenly into all the terms in an expression. For example,

$$4x^2+8x-12x^3$$

We have three terms, they all have a coefficient that is a multiple of 4, and a common factor of all these coefficients is 4.

$$4(x^2+2x-3x^3)$$

We can also take out an x from each of the three terms.

$$4x(x+2-3x^2)$$

So 4x is the highest common factor, there is nothing left that is common to take out from the three terms.

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Here's another way of looking about it, we can write all factors of each term and identify the largest subset of what's common between them :

$$\begin{align*} 4x^2 &: 4 \cdot x \cdot x \\ 8x &: 4 \cdot 2 \cdot x \\ -12x^3 &: 4 \cdot -3 \cdot x \cdot x \cdot x \end{align*}$$

Now identify what's common :

$$\begin{align*} 4x^2 &: \colorbox{yellow}{4} \cdot \colorbox{yellow}{$x$} \cdot x \\ 8x &: \colorbox{yellow}{4} \cdot 2 \cdot \colorbox{yellow}{$x$} \\ -12x^3 &: \colorbox{yellow}{$4$} \cdot -3 \cdot \colorbox{yellow}{$x$} \cdot x \cdot x \end{align*}$$

All terms share a HCF of 4x.