How to Factor Binomials That Are Not Perfect Squares
Binomial expressions are a fundamental concept in algebra, and factoring them is a crucial skill for students to master. While factoring perfect square binomials can be straightforward, the process becomes more complex when dealing with binomials that are not perfect squares. In this article, we will explore various methods to factor binomials that are not perfect squares, helping you become a proficient algebraic factorizer.
1. Find the Greatest Common Factor (GCF)
The first step in factoring binomials that are not perfect squares is to identify the greatest common factor (GCF) of the terms. The GCF is the largest number that divides both terms without leaving a remainder. Once you have found the GCF, factor it out from both terms.
For example, consider the binomial expression \(6x^2 + 12x\). The GCF of 6x^2 and 12x is 6x. Factor out 6x from both terms:
\(6x^2 + 12x = 6x(x + 2)\)
Now, you have factored the binomial expression, and the resulting factors are 6x and (x + 2).
2. Factor by Grouping
Another method to factor binomials that are not perfect squares is by grouping. Group the terms in pairs and factor out the GCF from each pair. Then, factor out the GCF of the two resulting binomials.
For instance, let’s factor the binomial expression \(9x^2 – 3x – 6x + 2\). Group the terms in pairs:
\(9x^2 – 3x – 6x + 2 = (9x^2 – 3x) – (6x – 2)\)
Factor out the GCF from each pair:
\(9x^2 – 3x – 6x + 2 = 3x(3x – 1) – 2(3x – 1)\)
Now, factor out the GCF of the two resulting binomials:
\(9x^2 – 3x – 6x + 2 = (3x – 1)(3x – 2)\)
The binomial expression has been factored successfully.
3. Use the Distributive Property
The distributive property is another tool that can be used to factor binomials that are not perfect squares. By multiplying the binomial by a form of 1, you can create a perfect square trinomial or a difference of squares, which can then be factored.
For example, consider the binomial expression \(2x^2 – 5x + 2\). To factor this expression, we can multiply it by a form of 1, such as \((2x – 1)(2x – 1)\):
\(2x^2 – 5x + 2 = (2x – 1)(2x – 1)\)
Now, expand the expression:
\(2x^2 – 5x + 2 = 4x^2 – 4x + 1\)
The resulting expression is a perfect square trinomial, which can be factored as:
\(4x^2 – 4x + 1 = (2x – 1)^2\)
So, the original binomial expression can be factored as:
\(2x^2 – 5x + 2 = (2x – 1)^2\)
4. Practice and Review
To become proficient at factoring binomials that are not perfect squares, it is essential to practice and review the different methods discussed in this article. By solving various problems and applying these techniques, you will develop a deeper understanding of the process and become more confident in your algebraic skills.
In conclusion, factoring binomials that are not perfect squares can be challenging, but with the right techniques and practice, you can master this skill. By following the methods outlined in this article, you will be well on your way to becoming an expert in factoring binomials of all types.
