## Trinomials

An expression , where , and are constants and , is called a **quadratic trinomial** in . It has three terms which are an term, an term and an independent term.

In equations, these terms may also be written as **quadratic equations**. Factorising these quadratic trinomials/equations yield a **binomial** product.

There are two types of quadratic trinomials:

*1. Monic Trinomials*

"Monic" means the first coefficient is 1, for example .

Factorising monic trinomials is a simple operation. Using the example above, the factors could be obtained by **finding two numbers with sum 16 and product 48**. Notice that 16 is the second coefficient and 48 is the independent number from the quadratic equation.

Hence, the numbers are 12 and 4 since 12 + 4 = 16 and 12 x 4 = 48.

Then we put them in the following format:

Hence,

There are many trial-and-error attempts in order to factorise these equations.

### Examples:

*2. Non-Monic Trinomials*

"Non-monic" means the first coefficient is . There are several effective ways of factorising these non-monic quadratic trinomials: cross method, PSF (Product and Sum Factor) and quadratic formula, which is discussed in the Quadratic Formula section.

#### Cross Method:

For example, factorise .

List down factors of and .

Factors of are and

Factors of 2 are 1 and 2.

From these factors, a trial and error pairing process is conducted until it fits the original equation.

Place the factors one above the other as shown below:

As we multiply by 2 and by 1, we obtain the sum of . However, the sum is not as expected from the original equation's second term. Hence, another arrangement needs to be tested out.

Now, the sum is , as expected from the equation. Hence, this arrangement could be accepted.

(Example courtesy of mathsteachers.com.au)

#### Product and Sum Factor:

Watch this video to understand the factorising of quadratic trinomials using Product and Sum Factor.