Factor a trinomial, special cases

Trinomials of the form ax^2+bx+c where a\ne1

A trinomial where a\ne 1 can be difficult to factor. In grade 11 pre-calc, you are required to factor any trinomial of the form ax^2+bx+c. In grade 10, we need only consider three special cases.

Remember that not all trinomials factor nicely with integers. All the examples here are chosen because they do factor nicely with integers.

Special case 1: No constant term (officially a binomial…).

    \[4x^2-16x\]

The highest common factor of these two terms is 4x.

    \begin{align*}&4x^2-16x\\[10pt]=&4x(x-4)\end{align}

Special case 2: A common factor between all three terms

    \[2x^2-10x+8\]

First we notice that 2 divides all three terms. Let’s factor it out:

    \begin{align*}&2x^2-10x+8\\[10pt]=&2(x^2-5x+4)\end{align}

The trinomial in the brackets has a=1 and so we can factor by finding two numbers that multiply to 4 and add to -5.

    \begin{align*}&2x^2-10x+8\\[10pt]=&2(x^2-5x+4)\\[10pt]=&2(x-4)(x-1)\end{align}

Special case 3: A difference of two squares (also, officially a binomial, not a trinomial).

    \[4x^2-25\]

This example follows the format a^2 - b^2 = (a-b)(a+b)

In this case, a=2x and b=5. Therefore,

    \begin{align*}&4x^2-25\\[10pt]=&(2x-5)(2x+5)\end{align}

Practice:

Geogebra link

A note on names: A trinomial is a polynomial with three terms. A quadrilateral is a polynomial with degree 2, meaning that the highest power of a term is 2 and it has an x^2 term. Quadrilaterals are often trinomials, but not always.