Factoring Trinomials in Quadratic Form
If you know how to factor a quadratic expression, then you can factor a trinomial in quadratic form without issue.
On this page we will learn what a trinomial in quadratic form is, and what a trinomial in quadratic form is not. It’s really all about the exponents, you’ll see. Use the tabs below to navigate through the notes, video, and practice problems.
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There is one last factoring method you’ll need for this unit: Factoring quadratic form polynomials. A quadratic form polynomial is a polynomial of the following form:
Before getting into all of the ugly notation, let’s briefly review how to factor quadratic equations. Let’s consider two cases: (1) Leading coefficient is one, a = 1, and (2) leading coefficient is NOT 1, a ≠ 1.
If a is one, then we just need to find what two numbers have the product c and the sum of b. For example:
Here b = –2, and c = –15. To factor, we find a pair of numbers whose product is – 15 and whose sum is – 2. That would be a – 5 and a + 3.
For more practice on this technique, please visit this page.
If a is NOT one, things are slightly trickier. There are a lot of methods to factor these quadratic equations, but guess and check is perhaps the simplest and quickest once master, though mastery does take more practice than alternative methods.
Guess and check uses the factors of a and c as clues to the factorization of the quadratic. Here’s an example:
The first term, 2x2, comes from the product of the first terms of the binomials that multiply together to make this trinomial. Think FOIL. (2x + ?)(x + ?) = 2x2 + … The last term, – 5, comes from the L, the last terms of the polynomials. To get a -5, the factors are opposite signs. So either -5 × 1 or 5 × -1. To figure out which it is, just carry out the O + I from FOIL.
If you need a refresher on factoring quadratic equations, please visit this page.
Since factoring can be thought of as un-distributing, let’s see where one of these quadratic form trinomials comes from.
This is a quadratic form trinomial, it fits our form: 
Here n = 2.
Let’s factor a quadratic form trinomial where a = 1.
Step 1: Identify if the trinomial is in quadratic form.
Since (x2)2 = x4, and the second term is x4, then n = 2.
Remember, when a term with an exponent is squared, the exponent is multiplied by 2, the base is squared. So (3x5)2 = 9x10.
Step 2: Find the value of n.
As shown in step 1, the value of n is 2.
Step 3: Apply the appropriate factoring technique.
Let’s look at this quadratic form trinomial and a quadratic with the same coefficients side by side. This will help you see how the factoring works.
However, this quadratic form polynomial is not completely factored. Each factor is a difference of squares!
Let’s see one more example where a = 1.
Factor Completely:
This is a quadratic form polynomial because the second term’s variable, x3, squared is the first term’s variable, x6. So, n = 3.
The numbers that multiply to – 50 and add to + 5 are – 5 and + 10.
Non-Example: These trinomials are not examples of quadratic form.
Let’s see another example, here where a is not one.
Factor Completely:
This is a quadratic form trinomial because the last term is constant (not multiplied by x), and (x5)2 = x10. So, n = 5. The tricky part here is figuring out the factors of 8 and 30 that can be arranged to have a difference of 43.
Summary: A quadratic form trinomial is of the form axk + bxm + c, where 2m = k. It is possible that these expressions are factorable using techniques and methods appropriate for quadratic equations.
In other words, if you have a trinomial with a constant term, and the larger exponent is double of the first exponent, the trinomial is in quadratic form. It might be factorable.
