In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is x² − 4x + 7.
A polynomial isn't as complicated as it sounds, because it's just an algebraic expression with several terms. Usually, polynomials have more than one term, and each term can be a variable, a number, or some combination of variables and numbers.
Some people use polynomials in their heads every day without realizing it, while others do it more consciously. Since polynomials are used to describe curves of various types, people use them in the real world to graph curves.
For example, roller coaster designers may use polynomials to describe the curves in their rides. Combinations of polynomial functions are sometimes used in economics to do cost analyses, for example.
The daily life application is surprisingly very common. You may subconsciously use some polynomial when shopping! For example, you might want to know how much three pounds of flour, two dozen eggs, and three quarts of milk cost.
Before you check the prices, construct a simple polynomial, letting "f" denotes the price of flour, "e" denotes the price of a dozen eggs, and "m" the price of a quart of milk. It will look like this: 3f + 2e + 3m.
Therefore, learning about polynomials and how to factor polynomials are especially beneficial for you. In mathematics, factorization or factoring is the breaking apart of a polynomial into a product of other smaller polynomials.
If you choose, you could then multiply these factors together, and you should get the original polynomial (this is a great way to check yourself on your factoring skills).
You don’t have to worry much—when you have a polynomial, one way of solving it is to factor it into the product of two binomials. And it’s not as complicated as it may seem! Keep reading this article to learn how to factor polynomials.
What to know before learning how to factor polynomials
Before you start learning how to factor polynomials, it’s important to understand the concept of the distributive property to learn how to factor polynomials properly. For example, we can use the distributive property to find the product of 3x2 and 4x + 3 as shown below:
And because the distributive property is an equality, the reverse of this process is also true.
You also have to familiarize yourself with the concept of the GCF (greatest common factor). The GCF (greatest common factor) of two or more monomials is the product of all their common prime factors.
To factor the GCF out of a polynomial, find the GCF of all the terms in the polynomial, then express each term as a product of the GCF and another factor. Once done, use the distributive property to factor out the GCF.
How to factor polynomials
Now let’s learn how to factor polynomials from the basic, starting by factoring the GCF out of 2x3 – 6x2.
Step 1: Find the GCF
2x3 = 2 . x . x . x
6x2 = 2 . 3. x . x
So the GCF of 2x3 – 6x2 is 2 . x . x = 2x2.
Step 2: Express each term as a product of 2x2 and another factor.
2x3 = (2x2) (x)
6x2 = (2x2) (3)
So, the polynomial can be written as 2x3 – 6x2 = (2x2) (x) - (2x2) (3)
Step 3: Factor out the GCF
Now we can apply the distributive property to factor out 2x2. Look at the method below:
To check our factorization, simply multiply 2x2 squared back into the polynomial. If the result is equal to the original polynomial, our factorization is correct.
How to factor polynomials more efficiently
Once you’ve been familiar with the process of factoring out the GCF, you can use a faster method. This is because once we know the GCF, the factored form is simply the product of that GCF and the sum of the terms in the original polynomial divided by the GCF.
See, for example, how we use this fast method to factor 5x2 + 10x whose GCF is 5x: