Trinomials

A factorable trinomial will take the form ax^2+bx+cwhere 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

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