# Trinomials and Factorising

## Trinomials

An expression $a{ x }^{ 2 }+bx+c$, where $a$$b$ and $c$ are constants and $a\not=0$, is called a quadratic trinomial in $x$. It has three terms which are an ${x}^{2}$ term, an $x$ 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 ${x}^{2}+16x+48$.

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:

$(x +\qquad)(x +\qquad)$

Hence,

${x}^{2}+16x+48=(x+12)(x+4)$

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

### 2. Non-Monic Trinomials

"Non-monic" means the first coefficient is $\not=1$. 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 $7{x}^{2}+9x+2$.

List down factors of $7{ x }^{ 2 }$ and $2$.

Factors of $7{ x }^{ 2 }$ are $7x$ and $x$
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 $7x$ by 2 and $x$ by 1, we obtain the sum of $15x$. However, the sum is not $9x$ as expected from the original equation's second term. Hence, another arrangement needs to be tested out.

Now, the sum is $9x$, as expected from the equation. Hence, this arrangement could be accepted.

$\therefore 7{x}^{2}+9x+2 = (7x+2)(x+1)$

(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.