Factoring Polynomials

When it comes to solving polynomials, it is always handy to express them in a more concise way. That way, we can spot patterns that allow us to solve such polynomials easily. What if I told you that there is a nifty trick you can use to do just that? That's right! This nifty trick is called factoring. Let me give you a scenario. Say we are given the expression below,

$3x{y}^2-27x^3{y}^3+15x^{2}y-9x{y}^4$

Doesn't this look a little lengthy and somewhat confusing? However, do you see what each term has in common? Let me give you a hint: each term is divisible by 3, x and y. Taking these out of this expression, we will get a shorter and more straightforward expression. Though, more on that later!

In this article, we will be exploring factoring polynomials, their definition, how to solve polynomial equations and examples of applications of this.

Factoring

We will now begin with the definition of factoring.

This is an important topic when dealing with complex polynomials. Let us take the following example:

In algebra, factoring a (higher degree) polynomial means that we are rewriting a polynomial as a product of lower degree polynomials. Essentially, what we are doing here is the opposite of the FOIL method. Given the product of two binomials (a + b)(c + d), recall that the FOIL method is given by

Foil method, Aishah Amri - StudySmarter Originals

This is the result of expanding the product of the two binomials. We aim to simplify a polynomial so that it is displayed as a product of its lowest degree polynomials rather than multiplying it out.

Before we get into this topic, let us recall the following definitions.

Why is Factoring Important?

In most cases, factoring plays an important role in simplifying an expression. This allows you to solve a particular equation in a more efficient way. Furthermore, factoring will help you understand the behaviour of a polynomial expression when graphing is required. We can also solve equations through factoring polynomials by identifying their solutions. The following are the learning objectives for this section:

• Factoring a polynomial
• Solve polynomial equations via factoring

Here, we shall be introduced to three types of factoring methods, namely:

1. Greatest common factor (GCF)

2. Quadratic trinomials

3. Grouping

Greatest Common Factor

The Greatest Common Factor (GCF) is the highest common monomial shared between all the components of a polynomial. When factoring polynomials, it is important to start by using this method to avoid dealing with larger numbers.

To perform this method, we first seek the GCF and factor it out of the polynomial. Fundamentally, we are carrying out the reverse of the distributive law, or in notation form:

$ab+bc=b(a+c)$

The general case for factoring polynomials using the GCF is:

$a^3{b}^2+2a^{2}b-4a{b}^2=ab(a^{2}b+2a-4b)$

Notice that all the terms on the left-hand side of the general form above have the common factor ab. Below are some worked examples to demonstrate this factoring method:

Quadratic Trinomials

A second-degree polynomial is known as a quadratic. This means that the largest exponent in the polynomial is a 2. Typically, they take on the form of a quadratic trinomial as there are three terms, that is:

$ac{x}^2+(ad+bc)x+bd$

For this method of factoring, we seek to factor quadratic trinomials into a product of first-degree binomials. There are five steps to consider when applying this method:

Step 1: Write down a pair of parenthesis: ( ) ( )

Step 2: Deduce the product of the first term in the trinomial and add it into the parenthesis as: $\left(ax+\square \right)\left(cx+\square \right)$

Step 3: Look for factors of the third term of the trinomial. This should be in the form of a pair, such as $\left(b\right)\left(d\right)=bd$

Step 4: Use the trial and error method to identify which pair of factors satisfy the quadratic polynomial. In other words, find the pair that add up to the middle term of the trinomial. Remember to plug in all possible pairs and every possible order to determine the correct pair of values.

Step 5: Once found, add these values into the complete factorized form:$(ax+b)(cx+d)$

This form of factorization can be rather lengthy and complicated. Mastering this technique comes with lots of practice! Once you get the hang of the trial and error method, it will be very straightforward. We shall make use of the general formula below when factoring quadratic trinomials:

$ac{x}^2+(ad+bc)x+bd=(ax+b)(cx+d)$

Let us now look at some worked examples that exercise this factoring method.

Grouping

The grouping method is often used when we encounter a polynomial with four terms. There are three steps to this method:

Step 1: Group the polynomial into two sets of two terms. This is usually done by splitting the first two terms and the last two terms of the polynomial as,$(ax+bx)+(ay+by)$

Step 2: Factorize each group using the GCF factoring method: $x(a+b)+y(a+b)$

Step 3: If we find a common polynomial between the two factorized groups, apply the GCF factoring method again to factor it out: $(a+b)(x+y)$

The general case for factoring polynomials via grouping is as follows:

$ax+bx+ay+by=x(a+b)+y(a+b)=(a+b)(x+y)$

We shall now observe the following worked examples that employ this factoring method.

Grouping 3-Term Polynomials

Grouping is an amazing way to factor 3 term polynomials as well! Say we are given the expression below

$a+b+c$

Here, we shall use the trial and error method to find the 2 numbers that add to b and multiply to get ac. This makes a 4 term polynomial as below.

$a+\underset{=b}{\underbrace{aB+cB}}+c$

Finally, we shall apply the grouping method to simplify the expression as before

$a(1+B)+c(1+B)=(a+c)(1+B)$

Let us look at the example below.

Factoring Polynomials of Higher Degree

So far, the polynomials that we have dealt with have a degree of two. What if we need to factorize polynomials of degree higher than that?

When tackling such expressions, there is no particular method in factorizing them. However, we can indeed apply the techniques introduced in this lesson. It is helpful to become well acquainted with these factoring methods so that you can easily seek common patterns when solving them. Let us look at some worked examples to show this.

Solving Polynomial Equations

Now that we have mastered factoring polynomials, we can move on to solving them! Say we have the standard form of a polynomial equated to zero as below:

$ac{x}^2+(ad+bc)x+bd=0$

As before, we know that the factorized form for this polynomial is:

$(ax+b)(cx+d)=0$

Notice that the product of these two factors is zero if at least one of the factors is zero, that is either $(ax+b)=0or(cx+d)=0$. This is known as the Zero Product Property, stated below:

The Zero Product Property

If ab = 0 then a = 0 or b = 0 (or both a = 0, b = 0)

Applying this property, we can solve the general form of a factorized polynomial as above, by identifying that:

$ax+b=0\mathit{a}\mathit{n}\mathit{d}cx+d=0$

Rearranging this in terms of x, we have two solutions:

$x=-\frac{b}{a}\mathit{a}\mathit{n}\mathit{d}x=-\frac{d}{c}$

Do you see how factoring plays an important role in solving equations? Solving the equation via factoring provides the x-intercepts of the graph for a given equation. Let us return to the last three worked examples we did to show this.

It is helpful to note that when solving repeated factor equations, we only obtain one solution. For example,