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1.08 Factorising monic quadratic trinomials

Lesson

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

Worked Solution
Create a strategy

Find two numbers p and q that add to 6 and multiply to 8, then factorise using \left(x+p\right)\left(x+q\right).

Apply the idea

Notice that 6 and 8 are positive, then both p and q must be positive. We need factors of 8 have add to 6, which are 2 and 4.

\displaystyle x^2+6x+8\displaystyle =\displaystyle (x+2 )(x+4)Factorise

Example 2

Factorise: x^2-12x+36

Worked Solution
Create a strategy

Use the rule for perfect square expansion: A^2-2AB+B^2=\left(A-B\right)^2.

Apply the idea

There are two square terms, x^2 and 36=6^2. This means A=x and B=6. We know this works because 2AB=2\times 6\times x=12x.

\displaystyle x^2-12x+36\displaystyle =\displaystyle \left(x-6\right)^2Use the rule

Example 3

Factorise: x^2-17x+60

Worked Solution
Create a strategy

Find two numbers that add to -17 and multiply to give 60, then factorise.

Apply the idea

We need to choose factors of 60 that have sum of -17. The factors are -5 and -12.

\displaystyle x^2-17x+60\displaystyle =\displaystyle \left(x-5\right)\left(x-12\right)Factorise
Idea summary

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).

Outcomes

VCMNA332

Expand binomial products and factorise monic quadratic expressions using a variety of strategies.

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