Factorise monic quadratic trinomials
We call expressions of the form x^2+mx+n, where x is a pronumeral and m and n are numbers, monic quadratic trinomials. In order to factorise these, we want to use the rule AC+AD+BC+BD=\left(A+B\right)\left(C+D\right), but there are three terms instead of four.
The first term is x^2. Since x is a pronumeral, we can't really split it up, so to fit the distributive law we want A=x and C=x.
If we also let B=p and D=q, then we get \left(x+p\right)\left(x+q\right)=x^2+px+qx+pq by expansion. We can then factorise x from the two middle terms to get x^2+\left(p+q\right)x+pq.
Comparing this to the monic quadratic expression we have x^2+mx+n=x^2+\left(p+q\right)x+pq. Equating the coefficients of x tells us m=p+q and n=pq. This means that there are two numbers, p and q which add to give m and multiply to give n. If we can find these two numbers we can factorise the monic quadratic expression.
Examples
Example 1
Factorise: x^2+6x+8
Example 2
Factorise: x^2-12x+36
Example 3
Factorise: x^2-17x+60
An expression of the form x^2+mx+n is a monic quadratic trinomial.
To factorise expressions like this, we find a pair of numbers p and q such that p+q=m and pq=n.
Then the factorisation is x^2+mx+n=\left(x+p\right)\left(x+q\right).