Factor Trinomial

To factor a trinomial is to rewrite the sum of three terms

    \[x^2+7x+12\]

as an expression that is the product of two terms:

    \[(x+3)(x+4)\]

Straight to examples

Factoring is used for solving quadratic equations in any context. In school factoring helps to draw and understand graphs of quadratic functions. Factoring is one of several methods of solving a quadratic equation – not all quadratic expressions factor but if they do, solving equations by factoring can be the fastest method.

If a particular quadratic expression doesn’t factor easily, we can use alternative methods for solving equations/graphing functions.

In addition to solving equations and graphing, factoring allows us to use a different form of the same expression. It can be helpful when proving results in other areas of mathematics, such as vector geometry and calculus.

Checking your work

The first skill required to factor a trinomial is to be able to expand the brackets to check that the three terms compute correctly.

The puzzle

The second skill required is to be able to solve a numerical puzzle: two numbers that add to one value and multiply to another – see the applet below.

applet link: https://www.geogebra.org/m/eUSZMepF

Examples

Video: from Khan Academy

Example:

    \[x^2+7x+10 =(x+a)(x+b)\]

We need to find numbers a and b such that

    \[a\times b = 10 \quad \text{and}\quad a+b=7\]

A little thought or writing on paper leads us to the values 2 and 5 because 2\times 5 = 10 and 2 + 5 = 7.

Therefore we have

    \[x^2+7x+10= (x+2)(x+5)\]

Check this answer by expanding the brackets with FOIL or otherwise to confirm that the expression on the left hand side is equivalent to the expression on the right hand side.

The applet below is for practice factoring ax^2+bx+c where a=1. Note: all the trinomials given here factor nicely with integers. This is not always the case – chances are pretty high that if you make up a random trinomial, it won’t factor so nicely.

applet link


3. Factors and Products

Next: Factor a trinomial with other leading coefficients