Algebra has its many quirks. When you work with polynomials and factoring in algebra, you learn the tips and tricks that can make breaking down larger polynomials into smaller pieces easier. The method of factoring by grouping is one of those helpful methods.
The method of factor by grouping allows you to find common factors by focusing on smaller groups instead of on the larger polynomial. Working with smaller groups of terms instead of the polynomial allows you to see patterns more easily.
I know this is a mouthful, but once we start to look behind the concept, we’ll be able to see how all of this works.
What is factoring by grouping?
When we are asked to factor a polynomial, we usually have options, depending on what kind of polynomial is in front of us.
There are times when finding factors for larger polynomials is a bit difficult.
When I say larger polynomials, I refer to polynomials with powers higher than 2. Polynomials with smaller degrees have other options not available to these higher degree polynomials.
For instance, various other methods can be used when you can’t factor a quadratic. You can use the method of completing the squares, or you can use the quadratic formula to find the factors.
The method of factoring by grouping allows you to break the larger polynomial into groups to focus on terms within groups that have common factors. Once you’ve isolated those common factors within smaller groups, common factors start to show themselves.
Let’s look at the equation y=x3+3x2-x-3. It might be hard to figure out how to factor this down into smaller parts when you first look at it.
But what if we grouped terms together. When we split the polynomial into two different groups, we see common factors that we couldn’t see before. The x2 is a common factor in the first two terms.
As we factor each group, we see common factors between the groups. The x+3 becomes a common factor between the two groups.
I pulled that factor out from the two groups, leaving us with x2-1.
The factors for this polynomial are (x+3) and (x2-1). The x2-1 should be easy to factor.
When we factor by grouping, we are factoring the polynomial on different levels. By grouping the terms, we take on a few groups at a time instead of the whole thing.
Factor by grouping on trinomials
This might not be the ideal way to factor a trinomial or a quadratic equation, but it is helpful to know. And once you learn how to perform this method on these equations, performing it on larger polynomials will be a piece of cake.
Here is what you need to know.
To perform this method on a quadratic equation, we have to have four terms. And the way that we do this is by focusing on the middle x term and its coefficient.
Before we look at a true three-term quadratic equation, we’ll look at a quadratic equation with four terms. The middle term has been split into two terms.
Let’s take a look at a simple example. The equation is y=x2-12x+8x-96.
This equation has four terms, so we should be able to do factoring by grouping them. We group together the first two terms and then the last two terms, and then we factor out common factors in each group.
Once the common factors in each group have been factored out, we can then see a common factor between each grouping.
The equation y=x2-12x+8x-96 is equal to the equation y=x2-4x-96. I’m sure you’re asking what the process of splitting up that middle term is to create two terms?
When we have a quadratic equation, y=ax2+bx+c, we must look at the first coefficient, a, and the constant, c. We multiply these two terms and set them aside.
In the example, a=1 and c=-96. For the product, we would get 1* -96, or -96.
We then consider all the possible factors for -96 and find the values that add to the middle term -4.
In the case of the example, the factors that we need are 8 and 12, but we need them to total the value of -4. To do that, we need to have (-12 + 8).
Our middle term splits into the terms -12x + 8x. Once we have that middle term split, we can group terms and pull out common factors.
Let’s try a few more examples.
The following equation is y=2x2-5x-12.
The first step is to figure out what a*c is. In our equation, a=2 and c=-12. When we multiply them out, we get -24. We will look for factors of -24 that, when summed, will equal the middle term, -5.
In this case, the factors that work best are -8 and 3. Because the constant is negative, we know that we are looking for one factor to be positive and one to be negative. Since the middle term is negative, we know that the larger factor has to be negative.
We split the middle term into two terms: 8x and 3x. When multiplied out, their product equals -24, and when summed, their total is equal to -5.
The next step is to group terms into smaller groups of two to find factors in each smaller group. The common factor in the first group is 2x. the common factor in the second group is 3.
We can pull that value from each term once we see each group’s common factor, x-4.
We can finish factoring this equation now that its pieces are more manageable.
The next example is y=32x2-46x+15.
Going through the same process, we take the coefficients a and c and multiply them out in this equation a=32 and c=15. The total comes to 480.
We then start to look at the factors that, when added together, will add up to the middle term.
Remember that because the constant is positive, but the middle term is negative, both factors will need to be negative.
After much trial and error, the two values are found. They are -16 and -30. These two values multiply out to 480. When summed, they equal -46.
The middle term is split into the terms -16x and -30x.
We then group the first two terms together and pull out the common factor (16x). We do the same for the second two terms (-15).
Once those values are pulled out, we see the common factor, 2x-1, between the two groupings.
We can pull that factor from the first grouping and the second grouping. The equation we are left with is more manageable than what we started with.
I know that there are other ways to factor a quadratic. Sometimes those ways are easier. This is just another method to store away when you need a helping hand.
Factor by grouping with polynomials with four or more terms.
Factor by grouping is useful on polynomials that have four terms or more. It’s also more helpful to use factoring by grouping on a polynomial with an even number of terms.
For larger polynomials with an odd number of terms, other methods might be more useful than factor by grouping. For instance, if you have a complex polynomial that needs to be factored, using rational root theorem and synthetic division together might be more helpful.
Let’s try this method on the equation, y=x4+3x3-8x-24. Because there are already an even number of terms, it makes it easy to group the terms for factoring.
In this equation, we look for common factors between groups of terms. There is a common factor in the first group (x3) and a common factor in the second group (-8).
When those common factors are pulled from each grouping, we find a common factor, x+3, between each grouping.
We pull the common factor from each grouping and then regroup terms. What we are left with is (x+3)(x3-8).
The last example we’ll look at is y=x3-7x2-49x+343.
We group terms together that have common factors. We group the first two terms and the last two terms in this case.
The common factor in the first grouping is x2. The common factor in the second grouping is -49.
By doing this, we can see the common factor, x-7, between the two groupings.
We pull this factor from each grouping which leaves us with (x-7)(x2-49). This is way easier to deal with than what we had before.
Never forget that these tools are meant to help you. Learn to master them so that you can be successful.
Stay consistent and work hard. You’ve got this! 😁