Video: Factorization Using the Highest Common Factor

In this video, we will learn how to factor algebraic expressions by identifying the highest common factor (HCF).

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Video Transcript

In this video, we will learn how to factor algebraic expressions by identifying the highest common factor. Before we talk about the highest common factor, let’s remember what factors are. We can write numbers as a product of their factors. For example, if we have 12, we could write it as two times six. So, we say two and six are factors of 12. But we also know that six equals two times three, which means we could say that 12 equals two times two times three. Let’s do this again with the number 18. Two times nine equals 18. And three times three equals nine. And so, we could say that 18 equals two times two [three] times three.

And when we’re comparing two numbers, sometimes it’s helpful to identify the common factors. And the common factors are the factors shared by both of the numbers. In this case, 12 and 18 both have a factor of two and a factor of three. These are common factors. But often when we’re comparing numbers, we won’t be interested in the common factors. We’ll be interested in the highest common factor. You might see that abbreviated as the HCF. You might also see this referred to as the greatest common factor or the GCF. The greatest common factor or the highest common factor will be the largest whole number that is a factor of both. It will also be the product of all the prime common factors.

Here, we see 12 equals two times six, and 18 equals three times six. Six is the largest whole number factor of both of these values. And so, we would say that the HCF of 12 and 18 is six. But what if we’re trying to compare these two values, two 𝑥 squared 𝑦 and four 𝑥𝑦? What is the highest common factor here? First, let’s break apart the factors. We could say that this equals two times 𝑥 squared 𝑦. And the factors of 𝑥 squared 𝑦 are 𝑥 squared and 𝑦. 𝑥 squared has two factors, 𝑥 and 𝑥. And so, two 𝑥 squared 𝑦 can be broken down into its factors, two times 𝑥 times 𝑥 times 𝑦. And we do the same thing for four 𝑥𝑦, four times 𝑥𝑦. The four can be broken up into two factors of two. And the 𝑥𝑦 is 𝑥 times 𝑦, which means we have four 𝑥𝑦 equals two times two times 𝑥 times 𝑦.

Both of these values have factors of two, 𝑥, and 𝑦. And so, the highest common factor is two 𝑥𝑦. We use the highest common factor to simplify certain expressions. But before we look at that, there’s one more property we need to remember, and that’s the distributive property. It tells us that 𝑎 times 𝑏 plus 𝑐 is equal to 𝑎 times 𝑏 plus 𝑎 times 𝑐. We could say that the 𝑎 here is the highest common factor of the values 𝑎 times 𝑏 and 𝑎 times 𝑐. Simplifying using the highest common factor is then undistributing the highest common factor from the terms. So, let’s look at the example that was on the opening screen. Using the highest common factor, we’ll simplify the expression.

Using the diagram, factor four 𝑥 plus 12.

This diagram has images that reflect four 𝑥 and 12. The diagram on the right-hand side has combined the two images. And it’s taken the bar of size 12 and divided it into four evenly sized pieces. We know that 12 divided by four equals three. This diagram has removed a factor of four from the 12 and from the four 𝑥. 12 divided by four equals three. And four 𝑥 divided by four equals 𝑥.

We see that four times 𝑥 plus four times three equals four 𝑥 plus 12. These are equivalent expressions. And so, we can say that four 𝑥 plus 12 is equal to four times 𝑥 plus three. We can also say that the highest common factor of four 𝑥 and 12 is four. We have undistributed the highest common factor from these two terms to give us four times 𝑥 plus three.

Here is another example.

Factor 15𝑒 plus 15𝑓 completely.

We’re given the expression 15𝑒 plus 15𝑓. We know that 15𝑒 equals 15 times 𝑒, and we know that 15𝑓 equals 15 times 𝑓. Both values have a common factor of 15, which means 15 is the highest common factor. And if we undistribute this 15, we can write 15𝑒 plus 15𝑓 like this. 15 times 𝑒 plus 𝑓. We know that this is true based on the distributive property, which tells us 𝑎 times 𝑏 plus 𝑐 is equal to 𝑎 times 𝑏 plus 𝑎 times 𝑐. Since we’re not given any other information, this expression is factored as 15 times 𝑒 plus 𝑓.

Let’s look at an example where we’re finding the highest common factor of two terms that have variables.

Find the highest common factor of the two terms in this expression, four 𝑥 to the fourth power minus 18𝑥 cubed.

Our expression is four 𝑥 to the fourth power minus 18𝑥 cubed. And here are the two terms. We’re trying to find the highest common factor. The first term has a factor of four and a factor of 𝑥 to the fourth power. The second term has a factor of 18 and 𝑥 cubed. But four is not a factor of 18. But we recognize four and 18 are both even numbers, so we know they both have a factor of two. Four is two times two, and 18 is two times nine. So, both have a factor of two.

But now we need to think about how we would deal with this 𝑥 to the fourth power and 𝑥 cubed. We could say that 𝑥 to the fourth power is equal to 𝑥 to the first power times 𝑥 cubed. Then, we see that both of these terms have a factor of 𝑥 cubed. So, we can rewrite four times 𝑥 to the fourth power as two times 𝑥 cubed times two 𝑥. And 18𝑥 cubed can be rewritten as two 𝑥 cubed times nine, which shows that two 𝑥 cubed is the highest common factor of these two terms.

So far, we’ve looked at comparing two numbers or expressions that only have two terms. We’re now going to find the highest common factor of an expression that has three terms. And the process will be the same no matter how many terms an expression has.

Factor the expression six 𝑝 squared plus three 𝑝 minus six 𝑝𝑞 completely.

Given the expression six 𝑝 squared plus three 𝑝 minus six 𝑝𝑞, we need to find the highest common factor. The coefficients of all three terms are divisible by three. We know that we could then undistribute a three. For the first term, six 𝑝 squared would be equal to three times two 𝑝 squared. For the second term, if we remove a factor of three, we’ll be left with 𝑝 because three times 𝑝 equals three 𝑝. For the third term, we’ll have three times negative two 𝑝𝑞 because three times negative two 𝑝𝑞 equals negative six 𝑝𝑞.

However, we haven’t yet removed the highest common factor. We know this because we see a factor that still remains in all three terms. All three terms have at least one factor of 𝑝. Now, we want to undistribute this factor of 𝑝, that is, a factor of 𝑝 to the first power. To remove a factor of 𝑝 from the first term, we’ll be left with two 𝑝. Now, the middle term is the trickiest. To remove a factor of 𝑝, we need to think 𝑝 to the first power times what equals 𝑝 to the first power. And that would be one. 𝑝 divided by 𝑝 equals one.

And finally, to remove a factor of 𝑝 from negative two 𝑝𝑞, we would be left with negative two 𝑞, which means we have three 𝑝 times two 𝑝 plus one minus two 𝑞 as our factorized expression. If we wanted to check and see if this was true, we would redistribute the three 𝑝 across all three terms. Three 𝑝 times two 𝑝 equals six 𝑝 squared. Three 𝑝 times one equals three 𝑝. And three 𝑝 times negative two 𝑞 equals negative six 𝑝𝑞. This is the expression we started with, and so we found the factored form. Three 𝑝 times two 𝑝 plus one minus two 𝑞.

Let’s consider another example.

Factor fully 𝑎 minus 10 times 𝑎 plus eight minus two times 𝑎 plus eight.

Given the expression 𝑎 minus 10 times 𝑎 plus eight minus two times 𝑎 plus eight, in order to factor this, we need a common factor between the two terms. Here are the two terms. The first term has a factor of 𝑎 minus 10 and a factor of 𝑎 plus eight. And the second term has the factors negative two and 𝑎 plus eight, which means both terms share a factor of 𝑎 plus eight. And that means we can undistribute the factor of 𝑎 plus eight. In our first term, if we take out the factor 𝑎 plus eight, the factor remaining will be 𝑎 minus 10.

In our second term, if we remove 𝑎 plus eight, we’ll be left with negative two. We’ve now rewritten our original expression as 𝑎 plus eight times 𝑎 minus 10 minus two. And within these brackets, we can do some simplification. Since there’s only addition or subtraction inside the brackets, we can remove the parentheses. So, we have 𝑎 plus eight times 𝑎 minus 10 minus two. And 𝑎 minus 10 minus two equals 𝑎 minus 12. A fully factorized form of the original expression would look like this. 𝑎 plus eight times 𝑎 minus 12.

In our final example, we’ll look at another expression with multiple variables.

By taking out the HCF, factor the expression 14𝑥 to the fifth power 𝑦 squared minus four 𝑥 cubed 𝑦 plus eight 𝑥 squared 𝑦.

We’re given the expression 14𝑥 to the fifth power 𝑦 squared minus four 𝑥 cubed 𝑦 plus eight 𝑥 squared 𝑦, and we need the HCF, the highest common factor. We’ll start by considering the highest common factor of the coefficients of these three terms. 14 equals two times seven. Four equals two times two. And eight equals two times four. The common factor here is two. It is true that eight and four share a factor of four, but 14 is not divisible by four. And so, we say that the common factor for all three coefficients is two. And that means we’ll rewrite the expression as two times seven 𝑥 to the fifth power 𝑦 squared minus two 𝑥 cubed 𝑦 plus four 𝑥 squared 𝑦.

From here, we consider the common factor of 𝑥. The third term has the smallest factor of 𝑥, 𝑥 squared, which means 𝑥 squared is the biggest factor of 𝑥 we can remove. So, we’ll undistribute a factor of 𝑥 squared from these three terms. 𝑥 to the fifth power divided by 𝑥 squared equals 𝑥 cubed. And we leave the 𝑦 squared. 𝑥 cubed divided by 𝑥 squared equals 𝑥 to the first power. And four 𝑥 squared 𝑦 divided by 𝑥 squared will be equal to four 𝑦. We now have a second equivalent expression. However, we still do not have our highest common factor. And we know this because all three of our terms have at least one factor of 𝑦. The smallest factor of 𝑦 here is 𝑦 to the first power. And that means that’s the most we can remove from all three terms. We undistribute 𝑦 to the first power from all three terms.

Our first term becomes seven 𝑥 cubed 𝑦 to the first power. Our second term is then negative two 𝑥 to the first power. When we remove a factor of 𝑦 to the first power from our third term, we’re just left with four. What we see now is that there are no common factors in the parentheses. And that means the highest common factor is what we’ve taken out. By undistributing the highest common factor two 𝑥 squared 𝑦, we have a fully factorized expression, two 𝑥 squared 𝑦 times seven 𝑥 cubed 𝑦 minus two 𝑥 plus four. If you wanted to check if this was true, you would multiply the highest common factor back by the three remaining terms, which would give you the expression you started with.

Summarizing what we’ve seen, the highest common factor, HCF, or the greatest common factor, GCF, is the greatest shared factor when comparing the factors of two or more numbers. We use the distributive property which tells us that 𝑎 times 𝑏 plus 𝑐 is equal to 𝑎 times 𝑏 plus 𝑎 times 𝑐. And we can factor out the HCF using the distributive property to simplify expressions.

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