## Trinomials

A factorable trinomial will take the form $ax^2+bx+c$where $a$, $b$ and $c$ are integers.

There are two different kinds of trinomials that require similar techniques but one method is easier and the other is harder.

If $a=1$, then we refer to it as monic or an “easy trinomial” because the factoring method is easier.

If $a\neq 1$, then we refer to it as non-monic or an “hard trinomial” because the factoring method is more difficult.

#### Easy Trinomials

##### How to Factor an Easy Trinomial

Make sure that the trinomial is in standard form: $ax^2+bx+c$ and that $a=1$.

Then, find two numbers that multiply to $c$ and add to $b$.

The easy trinomial will factor to be:

$(x+$ first number $)(x+$ second number $)$

For example: $$x^2+7x+12$$

In this case, we must find two numbers that multiply to $c=12$ and add to $b=7$. The two numbers would be $3$ and $4$. So: $$x^2+7x+12=(x+3)(x+4)$$

#### Hard Trinomials

##### How to Factor a Hard Trinomial

Make sure that the trinomial is in standard form: $ax^2+bx+c$ and that $a$ is positive.

A hard trinomial is more difficult to factor than an easy trinomial. There are many different techniques.

One method is called “decomposition” which involves splitting the middle ‘$bx$‘ into two terms with new coefficients. These two coefficients must multiply to $a \times c$ and add to $b$.

You’ll then be able to “group/triple common factor” the four terms.

For example: $$3x^2-8x+4$$

In this case, we must find two numbers that multiply to $a \times c=12$ and add to $b=-8$. The two numbers would be $-2$ and $-6$. So, we split the $-8x$ term to be $-2x$ and $-6x$, then group: $$\begin{array}{rl} & 3x^2-2x-6x+4 \\ = & x(3x-2)-2(3x-2) \\ = & (3x-2)(x-2) \end{array}$$

#### Perfect Square Trinomials

A perfect square trinomial can be factored as a regular trinomial but there is a faster way to factor this special type. If the first term and last term are perfect squares, there’s a good chance you have a perfect square trinomial.

##### How to Factor a Perfect Square Trinomial

First, test to see if you have a perfect square trinomial. Take the square root of the first term and square root of the last term. If: $$|middle \; term| =2 \times \sqrt{first \; term} \times \sqrt{last \; term}$$

Then, you have a perfect square trinomial. It factors to: $$\Big( \sqrt{first \; term} + sign \; of \; b + \sqrt{last \; term} \Big)^2$$

For example: $$16x^2-24x+9$$

Notice that: $\sqrt{16x^2}=4x$ and $/sqrt{9}=3$. Also, $|-24x|=2(4x)(3)$. Therefore, it is a perfect square trinomial and: $$16x^2-24x+9=(4x-3)^2$$

### Practice

EXAMPLE 1

Factor $2x^2-10x+12$

###### SOLUTION

Using the Law of Division: $$2^{11} div 2^8=2^{11-8}=2^3$$

Evaluating: $2^3=8$

EXAMPLE 2

Factor: $4x^2-20xy+25y^2$

###### SOLUTION

Using the Law of Powers: $$(m^0)^{10}=m^0$$

We should more properly use the Law of Zero Exponents to write: $m^0=1$

EXAMPLE 3

Factor: $4x^2-5xy-6y^2$

###### SOLUTION

Since we are multiplying the two powers, we use the Law of Multiplication: $$x+4=7$$

Therefore: $x=3$