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Stage 5.1-3

2.12 Non-monic quadratic trinomials

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Non-monic quadratic trinomials

We call expressions of the form lx^2+mx+n, where x is a pronumeral and l, m and n are numbers, non-monic quadratic trinomials. Unlike monic quadratic trinomials, the coefficient of x^2 is not 1. Consequently we need to change our approach to account for this.

If we say that the factors are px+q and rx+s, then we can expand to get \left(px+q\right)\left(rx+s\right)=prx^2+\left(ps+qr\right)x+qs=lx^2+mx+n. Equating the coefficients gives us three equations: pr=l, ps+qr=m, and qs=n. Finding these four numbers will allow us to factorise the expression.

Examples

Example 1

Factorise: 6x^{2}-26x+24

Worked Solution
Create a strategy

Find factors of 6 and 24 to factorise in the form:

This image shows how to factorise a trinomial using the factors of the numbers. Ask your teacher for more information.
Apply the idea

We will find the combination of four numbers that make the outer and inner products add up to the middle term -26x.

This image shows how to factorise a trinomial using the outer and inner products. Ask your teacher for more information.

The factors of 6 are 2 and 3. The middle term -26x has a negative coefficient, we will need to have a pair of negatives in our factorisation that have a positive product of 24. These factors are -2 and -12.

By placing the numbers in the correct positions as above, we get the factorisation:

\displaystyle 6x^{2}-26x+24\displaystyle =\displaystyle 2\left(3x-4\right)\left(x-3\right)
Reflect and check

We can check our factorisation by expanding to make sure we get the original expression:

\displaystyle 2\left(3x-4\right)\left(x-3\right)\displaystyle =\displaystyle 2(3x^2-13x+12)Expand the brackets
\displaystyle =\displaystyle 6x^2-26x+24Simplify

Example 2

Factorise the expression by first taking out a common factor: 3x^2-21x-54

Worked Solution
Create a strategy

Take out a common factor of all the coefficients and constant term, then factorise the remaining quadratic.

Apply the idea

The common factor of 3, \, -21, \, -54 is 3. So we should take that out first.

\displaystyle 3x^2-21x-54\displaystyle =\displaystyle 3(x^2-7x-18)Take out the common factor

Now we can factorise the monic quadratic x^2-7x-18. -18 has factors of -9 and 2. -9+2=-7, so the factors are (x-9) and (x+2).

\displaystyle 3x^2-21x-54\displaystyle =\displaystyle 3(x-9)(x+2)Factorise the quadratic
Idea summary

An expression of the form lx^2+mx+n is a non-monic quadratic trinomial.

To factorise expressions like this, we find a numbers p, q, r, and s such that pr=l, ps+qr=m, and qs=n.

Then, the factorisation is lx^2+mx+n=\left(px+q\right)\left(rx+s\right).

Outcomes

MA5.3-5NA

selects and applies appropriate algebraic techniques to operate with algebraic expressions

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