In this video, we’re going to look at properties of operations and how they’re used for addition and multiplication, and then we will use these properties for factoring and expanding linear expressions and finally combining terms of an equation that are related to the factoring and expanding of linear expressions as well.
Now let’s look at some properties of addition and multiplication. So we have the associative property and here we have three plus five in parentheses plus seven equals three plus five plus seven.
So what this means is, it doesn’t really matter which two numbers the parentheses are around. And the same thing for multiplication, we have two times four in parentheses times six equals two times four times six.
Again it doesn’t matter which two numbers the parentheses are around. Now keep in mind these are just for addition and multiplication. They do not work for subtraction and division. The next property is the commutative property, and that’s just a property of the order.
So three plus five plus seven equals seven plus five plus three, and that would also equal five plus seven plus three. And for multiplication, two times four times six equals six times four times two, so it really doesn’t matter what order you put the numbers in. The solutions will be equal.
And then the distributive property works with a combination of addition and multiplication. So here we have two times four plus six. And what we do is, we distribute the two to the four and the two to the six. So we have two times four plus two times six, so that’s a combination of addition and multiplication.
So now we’ll use these properties to help with factoring and expanding and combining terms. Here we’re going to look at adding and subtracting linear expressions. So here’s an expression: four 𝑥 minus two plus six 𝑥 equals.
And when we add and subtract, what we want to do is combine like terms, so a like term could be just a number or in this case this term has an 𝑥 and this term has an 𝑥, so they’re called like terms. And so we can add the four 𝑥 and the six 𝑥 together to get ten 𝑥.
And this equals ten 𝑥 minus two because there’s only one constant term; we just do that. And there’s a minus sign here, so we carry that over here.
In this expression, we have seven plus three 𝑥𝑦 minus two 𝑥 plus eight equals. And notice here we have an 𝑥𝑦 term but only one of those and one 𝑥 term; even though they both have 𝑥s in them, 𝑥𝑦 is different from 𝑥, so they’re not like terms.
Here the like terms are the constants, the seven and the eight, so we combine those together and the linear expression becomes three 𝑥𝑦 minus two 𝑥 plus fifteen.
The last expression we’ll look at is ten 𝑦 squared plus six 𝑥𝑦 minus two 𝑦 squared plus five 𝑥 minus two. So again we have 𝑥𝑦 and 𝑥; those are different terms. But here the like terms we have a 𝑦 squared here and a 𝑦 squared here. So it’s ten 𝑦 squared minus two 𝑦 squared, which gives us eight 𝑦 squared.
And the expression is now eight 𝑦 squared plus six 𝑥𝑦 plus five 𝑥 minus two. So here we have the expression three 𝑥 times seven plus 𝑦. And remember, with the distributive property, we take this part outside the parentheses and we distribute it to both parts inside.
And we now have three 𝑥 times seven plus three 𝑥 times 𝑦, and this equals three 𝑥 times seven is twenty-one 𝑥 plus three 𝑥 times 𝑦 is three 𝑥𝑦.
Next we have four times 𝑢 minus one. So here’s the term outside we’re going to distribute to the terms inside the parentheses, which gives us four times 𝑢 plus four times negative one. And since four times negative one is negative four, this equals four times 𝑢 is four 𝑢 minus four.
And the last expression we’ll look at is negative five times six minus two 𝑚 plus three 𝑛. So in the first two expressions, we just had two terms inside the parentheses.
Here I put three just so you can see that it really doesn’t matter how many terms are inside the parentheses. We still take this term outside and we distribute to all the terms inside the parentheses.
And when we distribute the negative five, we have negative five times six plus negative five times negative two 𝑚 plus negative five times three 𝑛. And because of the negative term, negative five, we have to watch our signs here.
So this equals negative five times six is negative thirty. Now we have plus here; we have negative five times negative two 𝑚 becomes a positive ten 𝑚, so this is plus ten 𝑚, and then plus negative five times three 𝑛 which is negative fifteen 𝑛, so minus fifteen 𝑛.
And so here are some examples of how to use the distributive property to expand expressions. In this slide, we’re going to look at factoring expressions. In the last slide, we expanded expressions, so this is the opposite of that. And we factor expressions by looking at the terms.
So we have two terms here, four 𝑥 and six, and because only the first term has an 𝑥, we’re going to ignore the 𝑥, so we’re going to really look at the four and the six. And what we want to do is find the greatest common factor, so we see here that four and six are both divisible by two.
So the GCF, our greatest common factor, equals two. Now we take that two and we divide into each term. So if we have four 𝑥 divided by two, that equals two 𝑥, and six divided by two equals three.
So when we factor, we put the greatest common factor out front; that’s two times two 𝑥 plus three. And you can use what we learned in distributing the term outside the parentheses on the last slide to check and make sure that it equals four 𝑥 plus six.
Now let’s look at another expression. We have six 𝑥𝑦 minus twenty-seven 𝑥 squared. So first we look at the constant term here or the coefficient. We have six and twenty-seven; we’ll ignore this minus sign for now. And the greatest common factor of six and twenty-seven is three. Three divides into six as well as twenty-seven.
Now let’s look at the 𝑥 term; we have an 𝑥 here and we have an 𝑥 squared here. Remember when we just have a variaby- a variable by itself, the exponent is understood to be one.
And when we’re looking for the greatest common factor, we take the lower exponent and we factor that out. So here we have a one; here we have a two. So we add an 𝑥 to our greatest common factor. And here we have a 𝑦 term, but we don’t have a 𝑦 term in the second term here, so we’re only going to have a greatest common factor of three 𝑥, and we divide six 𝑥𝑦 by three 𝑥 and 𝑥 over 𝑥 cancel each other out; three divides into six two times, so this equals two 𝑦, and we take twenty-seven 𝑥 squared and we divide by three 𝑥, so three divides into twenty-seven nine times; this becomes one.
And here the 𝑥 disappears and the two as well. So this equals nine 𝑥. So now when we go and we write this, we have our factor three 𝑥 times two 𝑦, and here we have the minus, so we’re going to have the minus here nine 𝑥.
And again just like with the first expression, we can use the distributive property to check and make sure that when we distribute the three 𝑥 to the two 𝑦 minus nine 𝑥, we get six 𝑥𝑦 minus twenty-seven 𝑥.
So just to review here, we’ve learned how to add and subtract within linear expressions by combining like terms, we’ve learned how to distribute to expand the expression here, and we’ve learned how to factor by finding common terms within each term of the expression.