For more visual learners, feel free to watch a video of this same guide made by one of very own Instructors. You can follow along with the steps below!
If you’ve worked with polynomials before, you may be familiar with the Quadratic Formula. This formula is used to solve quadratic equations aka any equation where the highest degree is 2. If you have a polynomial with a degree bigger than 2 however, the Quadratic Formula would not work. How, then, can we solve polynomials of higher degrees? By factoring!
As a reminder, factoring means breaking down an expression into the smallest pieces we can to help us solve an equation. For example, let’s look at the following equation:
x^3 + 6x^2 + 11x + 6 = 0
The factors of this polynomial are (x+1), (x+2), and (x+3) which means that the solutions of the equation are x = -1, x = -2, and x = -3. You can confirm this by graphing the equation and looking for the x-intercepts.
Hold up! How, exactly, did we figure out the factors of this equation?? Let’s jump into this now!
Factoring by Greatest Common Factor
There are many different ways you can factor polynomial equations. The first one is to look for the greatest common factor (GCF)! The GCF is the biggest factor that every term in an expression has in common.
Let’s look at the following expression 24x^3 + 36x^2 + 54x. Can you find a factor that all three terms have in common? Notice that each term can be divided by 6!
Each term also share one x so we can take out 6x and simplify the rest of the expression. Now, we are left with:
Remember that whenever factoring an expression, you can always check if your GCF is correct by redistributing it back into the expression.
What do we do if there isn’t a common factor among all terms of an expression? Sometimes we can still find common factors within some of the terms. To do this, we can try another factoring method called Factoring by Grouping.
Factoring by Grouping
Factoring by Grouping works by splitting your polynomial into pieces before looking for a common factor. Let’s look at an example.
Examine the following expression: 6x^4 + 12x^3 + 3x + 6. Notice that at first glance, there are no common factors of all four terms. However, we can group the first two terms together and the last two terms together and look for a GCF within each group.
In the first one, we can factor out a 6x^3 and in the second group we can factor out a 3. This leaves us with 6x^3(x + 2) + 3(x+2).
Now we have two terms that share a factor of x+2! So if we factor x+2 out, we get (x+2)(6x3+3)!
Do you notice anything that can be simplified? We can pull out a 3 from our second binomial giving us our completely simplified expression 3(x+2)(2x3 +3).
Example 3: make another example
There are also some theorems we can use to factor out polynomial expressions with certain patterns. The first one is called Difference of Squares.
Difference of Squares
We can use the Difference of Squares method whenever we see a polynomial in the form of a^2 - b^2 = 0. What do you get when you try to factor this out? You get (a+b)(a-b). This is true for all differences of squares!
How can this help us with factoring higher degree polynomials, though? Let’s take a look at an example.
In the expression 16x^4 - 9, can you find two perfect squares? You may notice that both terms are perfect squares where a = 4x^2 and b = 3.
Using what we saw before, this means that we can factor 16x4 - 9 to (a+b)(a-b) or (4x+3)(4x-3).
Let’s try another example. How would you start to factor 3x^3 - 27x? At first glance, this does not appear to have a difference of squares. However, if you first factor out a 3, you are left with 3x(x2 - 9) and some perfect squares where a = x and b = 3.
So, now you get 3x(x+3)(x-3).
Don’t forget that that you can turn these expressions into equations and solve by simply setting each turn equal to 0. In this example, we would get x = 0, 3, and -3 which we can see check by graphing as well.
Sum of Cubes:
Similar to the Difference of Squares you can also find the Sum or Difference of Cubes. Whenever you find a Sum of Cubes, you can factor a3+b3 to (a+b)(a2 - ab + b2).
Don’t believe me? Well, we can multiply the expanded form and check to see if the two expressions are the same.
After multiplying everything out you can see that we do indeed get a^3 + b^3! Let’s check this theorem out in an example.
How can we factor the expression x^3+8 ?
Well, first we need to find the cubes in our problems which would be a = x and b = 2. Now, using our formula (a+b)(a2 - ab + b2), we would get (x+2)(x2-2x+4)!
Now, how would you factor 27x^6 + 125? This one seems intimidating but we can break it up into steps. Let’s look at the first term.
27 can be represented as 3^3 and x^6 can be represented as (x2)3 (because when raising a power to a power, you multiply the exponents). Moving on to the second term, 125 can be represented by 53 which leaves us with a = 3x^2 and b = 5.
So now, think you can factor this expression? If your final answer is (3x^2 + 5)(9x^4 - 15x^2 + 25), you are absolutely correct!
Difference of Cubes
Lastly, we have the Difference of Cubes. The expression a^3-b^3 can be factored down to (a-b)(a^2+ab+b^2).
So, if you are given x^3 - 8, how would you factor the expression? Well first, we know that our a = x for the first term. And we know 8 can be represented as 2^3 so b = 2.
a = x b = 2
In that case, using the pattern from before, we can factor x^3-8 to (x-2)(x^2+2x+4).
One last bonus problem! How would you factor out 40c^3-5d^3? One place we can start is finding the GCF which in this problem would be 5 so we get 5(8c^3 - d^3).
Now, you may notice that we have a difference of cubes where a = 2c and b = d. So, finally, we get 5(2c-d)(4c2+2cd+d2).
In summary, there are many ways to factor a higher degree polynomial. Sometimes, you may even use a combination of methods like we did in the last example. Overall, when trying to figure out the factors of a high-degree polynomial, here are some questions to ask:
Is there a common factor among all terms? If so, what is the greatest common factor?
Are there common factors with different groups of the polynomial?
Can I find a difference of squares?
Are there any sums or differences of cubes?
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