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In this video, we will learn how to factor algebraic expressions by identifying the highest common factor.
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Before we talk about the highest common factor, let’s remember what factors are.
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We can write numbers as a product of their factors.
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For example, if we have 12, we could write it as two times six.
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So, we say two and six are factors of 12.
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But we also know that six equals two times three, which means we could say that 12 equals two times two times three.
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Let’s do this again with the number 18.
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Two times nine equals 18.
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And three times three equals nine.
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And so, we could say that 18 equals two times two [three] times three.
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And when we’re comparing two numbers, sometimes it’s helpful to identify the common factors.
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And the common factors are the factors shared by both of the numbers.
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In this case, 12 and 18 both have a factor of two and a factor of three.
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These are common factors.
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But often when we’re comparing numbers, we won’t be interested in the common factors.
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We’ll be interested in the highest common factor.
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You might see that abbreviated as the HCF.
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You might also see this referred to as the greatest common factor or the GCF.
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The greatest common factor or the highest common factor will be the largest whole number that is a factor of both.
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It will also be the product of all the prime common factors.
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Here, we see 12 equals two times six, and 18 equals three times six.
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Six is the largest whole number factor of both of these values.
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And so, we would say that the HCF of 12 and 18 is six.
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But what if we’re trying to compare these two values, two 𝑥 squared 𝑦 and four 𝑥𝑦?
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What is the highest common factor here?
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First, let’s break apart the factors.
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We could say that this equals two times 𝑥 squared 𝑦.
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And the factors of 𝑥 squared 𝑦 are 𝑥 squared and 𝑦.
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𝑥 squared has two factors, 𝑥 and 𝑥.
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And so, two 𝑥 squared 𝑦 can be broken down into its factors, two times 𝑥 times 𝑥 times 𝑦.
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And we do the same thing for four 𝑥𝑦, four times 𝑥𝑦.
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The four can be broken up into two factors of two.
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And the 𝑥𝑦 is 𝑥 times 𝑦, which means we have four 𝑥𝑦 equals two times two times 𝑥 times 𝑦.
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Both of these values have factors of two, 𝑥, and 𝑦.
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And so, the highest common factor is two 𝑥𝑦.
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We use the highest common factor to simplify certain expressions.
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But before we look at that, there’s one more property we need to remember, and that’s the distributive property.
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It tells us that 𝑎 times 𝑏 plus 𝑐 is equal to 𝑎 times 𝑏 plus 𝑎 times 𝑐.
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We could say that the 𝑎 here is the highest common factor of the values 𝑎 times 𝑏 and 𝑎 times 𝑐.
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Simplifying using the highest common factor is then undistributing the highest common factor from the terms.
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So, let’s look at the example that was on the opening screen.
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Using the highest common factor, we’ll simplify the expression.
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Using the diagram, factor four 𝑥 plus 12.
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This diagram has images that reflect four 𝑥 and 12.
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The diagram on the right-hand side has combined the two images.
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And it’s taken the bar of size 12 and divided it into four evenly sized pieces.
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We know that 12 divided by four equals three.
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This diagram has removed a factor of four from the 12 and from the four 𝑥.
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12 divided by four equals three.
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And four 𝑥 divided by four equals 𝑥.
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We see that four times 𝑥 plus four times three equals four 𝑥 plus 12.
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These are equivalent expressions.
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And so, we can say that four 𝑥 plus 12 is equal to four times 𝑥 plus three.
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We can also say that the highest common factor of four 𝑥 and 12 is four.
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We have undistributed the highest common factor from these two terms to give us four times 𝑥 plus three.
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Here is another example.
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Factor 15𝑒 plus 15𝑓 completely.
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We’re given the expression 15𝑒 plus 15𝑓.
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We know that 15𝑒 equals 15 times 𝑒, and we know that 15𝑓 equals 15 times 𝑓.
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Both values have a common factor of 15, which means 15 is the highest common factor.
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And if we undistribute this 15, we can write 15𝑒 plus 15𝑓 like this.
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15 times 𝑒 plus 𝑓.
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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.
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Since we’re not given any other information, this expression is factored as 15 times 𝑒 plus 𝑓.
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Let’s look at an example where we’re finding the highest common factor of two terms that have variables.
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Find the highest common factor of the two terms in this expression, four 𝑥 to the fourth power minus 18𝑥 cubed.
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Our expression is four 𝑥 to the fourth power minus 18𝑥 cubed.
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And here are the two terms.
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We’re trying to find the highest common factor.
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The first term has a factor of four and a factor of 𝑥 to the fourth power.
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The second term has a factor of 18 and 𝑥 cubed.
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But four is not a factor of 18.
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But we recognize four and 18 are both even numbers, so we know they both have a factor of two.
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Four is two times two, and 18 is two times nine.
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So, both have a factor of two.
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But now we need to think about how we would deal with this 𝑥 to the fourth power and 𝑥 cubed.
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We could say that 𝑥 to the fourth power is equal to 𝑥 to the first power times 𝑥 cubed.
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Then, we see that both of these terms have a factor of 𝑥 cubed.
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So, we can rewrite four times 𝑥 to the fourth power as two times 𝑥 cubed times two 𝑥.
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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.
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So far, we’ve looked at comparing two numbers or expressions that only have two terms.
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We’re now going to find the highest common factor of an expression that has three terms.
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And the process will be the same no matter how many terms an expression has.
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Factor the expression six 𝑝 squared plus three 𝑝 minus six 𝑝𝑞 completely.
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Given the expression six 𝑝 squared plus three 𝑝 minus six 𝑝𝑞, we need to find the highest common factor.
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The coefficients of all three terms are divisible by three.
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We know that we could then undistribute a three.
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For the first term, six 𝑝 squared would be equal to three times two 𝑝 squared.
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For the second term, if we remove a factor of three, we’ll be left with 𝑝 because three times 𝑝 equals three 𝑝.
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For the third term, we’ll have three times negative two 𝑝𝑞 because three times negative two 𝑝𝑞 equals negative six 𝑝𝑞.
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However, we haven’t yet removed the highest common factor.
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We know this because we see a factor that still remains in all three terms.
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All three terms have at least one factor of 𝑝.
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Now, we want to undistribute this factor of 𝑝, that is, a factor of 𝑝 to the first power.
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To remove a factor of 𝑝 from the first term, we’ll be left with two 𝑝.
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Now, the middle term is the trickiest.
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To remove a factor of 𝑝, we need to think 𝑝 to the first power times what equals 𝑝 to the first power.
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And that would be one.
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𝑝 divided by 𝑝 equals one.
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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.
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If we wanted to check and see if this was true, we would redistribute the three 𝑝 across all three terms.
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Three 𝑝 times two 𝑝 equals six 𝑝 squared.
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Three 𝑝 times one equals three 𝑝.
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And three 𝑝 times negative two 𝑞 equals negative six 𝑝𝑞.
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This is the expression we started with, and so we found the factored form.
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Three 𝑝 times two 𝑝 plus one minus two 𝑞.
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Let’s consider another example.
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Factor fully 𝑎 minus 10 times 𝑎 plus eight minus two times 𝑎 plus eight.
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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.
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Here are the two terms.
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The first term has a factor of 𝑎 minus 10 and a factor of 𝑎 plus eight.
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And the second term has the factors negative two and 𝑎 plus eight, which means both terms share a factor of 𝑎 plus eight.
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And that means we can undistribute the factor of 𝑎 plus eight.
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In our first term, if we take out the factor 𝑎 plus eight, the factor remaining will be 𝑎 minus 10.
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In our second term, if we remove 𝑎 plus eight, we’ll be left with negative two.
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We’ve now rewritten our original expression as 𝑎 plus eight times 𝑎 minus 10 minus two.
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And within these brackets, we can do some simplification.
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Since there’s only addition or subtraction inside the brackets, we can remove the parentheses.
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So, we have 𝑎 plus eight times 𝑎 minus 10 minus two.
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And 𝑎 minus 10 minus two equals 𝑎 minus 12.
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A fully factorized form of the original expression would look like this.
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𝑎 plus eight times 𝑎 minus 12.
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In our final example, we’ll look at another expression with multiple variables.
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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.
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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.
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We’ll start by considering the highest common factor of the coefficients of these three terms.
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14 equals two times seven.
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Four equals two times two.
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And eight equals two times four.
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The common factor here is two.
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It is true that eight and four share a factor of four, but 14 is not divisible by four.
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And so, we say that the common factor for all three coefficients is two.
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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.
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From here, we consider the common factor of 𝑥.
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The third term has the smallest factor of 𝑥, 𝑥 squared, which means 𝑥 squared is the biggest factor of 𝑥 we can remove.
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So, we’ll undistribute a factor of 𝑥 squared from these three terms.
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𝑥 to the fifth power divided by 𝑥 squared equals 𝑥 cubed.
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And we leave the 𝑦 squared.
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𝑥 cubed divided by 𝑥 squared equals 𝑥 to the first power.
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And four 𝑥 squared 𝑦 divided by 𝑥 squared will be equal to four 𝑦.
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We now have a second equivalent expression.
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However, we still do not have our highest common factor.
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And we know this because all three of our terms have at least one factor of 𝑦.
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The smallest factor of 𝑦 here is 𝑦 to the first power.
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And that means that’s the most we can remove from all three terms.
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We undistribute 𝑦 to the first power from all three terms.
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Our first term becomes seven 𝑥 cubed 𝑦 to the first power.
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Our second term is then negative two 𝑥 to the first power.
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When we remove a factor of 𝑦 to the first power from our third term, we’re just left with four.
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What we see now is that there are no common factors in the parentheses.
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And that means the highest common factor is what we’ve taken out.
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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.
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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.
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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.
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We use the distributive property which tells us that 𝑎 times 𝑏 plus 𝑐 is equal to 𝑎 times 𝑏 plus 𝑎 times 𝑐.
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And we can factor out the HCF using the distributive property to simplify expressions.