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In this video, we'll be simplifying algebraic expressions. To begin with, we need to identify
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parts of an algebraic expression. A term is a number, a variable, or the product of a number
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and one or more variables raised to powers. These are examples of terms: negative four,
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x, negative four x, negative four x to the third power, six x cubed y squared, six x
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cubed over y squared, three-fifths a squared b c to the fourth. Each of these is a term.
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The coefficient of a term is the numerical factor of the term. Negative four is also a coefficient.
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The coefficient of this term is one. There is an implied one in front of the x. The
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coefficient of this term three-fifths a squared b c to the fourth is three-fifths.
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Terms with the same variable raised to the same powers are called like terms. If a term
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is not a like term, it's called an unlike term. Let's take a look at this example.
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Consider the term two-fifths a squared b c to the fourth power.
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What is the coefficient? In this term, the coefficient is two-fifths.
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Now think about what a like term might look like. A like term would have the same variables raised
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to the same power, but it could have a different numerical coefficient. So instead of two-fifths,
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perhaps it's three. Three a squared b c to the fourth would be an example of a like term.
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Any numerical coefficient in front of a squared b c to the fourth would work to create a like term.
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An unlike term could have any numerical coefficient, but it has to have either a
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different variable or a variable raised to a different power. So an example of an unlike term
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would be three a b c. Notice I have the same variables, but instead of an a to the second
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power, I have an a to the first power. And right away that makes it an unlike term.
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An algebraic expression containing the sum or difference of like terms
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can be simplified by applying the Distributive Property. This is called combining like terms.
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For example, three x plus two x. This can be rewritten using the Distributive Property as
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three plus two times x. Three plus two is five. This would give us five x. Let's do some examples.
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Simplify the expression by combining like terms. In our first example, we have negative y squared
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plus five y squared. Using the Distributive Property, we could rewrite this as negative
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one plus five times y squared. This would give us four times y squared, or four y squared.
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In the next example, we have fourteen a minus eight a plus seven. Let's identify where the like terms are. Fourteen
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a and negative eight a are like terms because they have the same variable raised to the same power.
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So using the Distributive Property, we can rewrite this as fourteen minus eight times a plus seven.
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This gives us fourteen minus eight which is six a plus seven. Since six a and seven are unlike terms, they cannot be
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combined. This is our simplified answer. In this next example, we have five a plus one plus four
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a minus nine. Let's identify our like terms. We have five a and four a which are like terms. Using the Commutative Property,
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I can rearrange this expression so that the five a and four a are next to each other.
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One and negative nine are also like terms. So let's use the Commutative Property. This
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is equal to five a plus four a plus one minus nine. And now using the Distributive Property,
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we have five plus four times a, and we can combine one minus nine to get negative eight.
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Five plus four is nine. So this gives us nine a minus eight. Nine a and minus eight are unlike terms,
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so this is our simplified answer. In the last expression, we have ten x plus three x squared
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plus two x minus x squared. Again, we begin by identifying like terms, the same variable
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raised to the same power. So even though there is an x here in this term and an x in this term,
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they are not like terms because they do not have the same power. Ten x and two x are like terms.
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I'll use the Commutative Property to put them next to each other, for clarity. And three x squared
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and minus x squared are also like terms. They have the same variable raised to the same power.
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Now using the Distributive Property, I have ten plus two times x
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plus three minus one times x squared, giving us a final answer of twelve x plus two x squared.
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When simplifying algebraic expressions, we often use the Distributive Property twice.
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Once, to remove the parentheses that are in the given expression and then again to combine like
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terms. Let's look at these examples. The directions are to simplify the expression.
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First, we'll use the Distributive Property to remove the parentheses.
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So negative four gets multiplied times three x and times the positive two, giving us
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negative twelve x minus eight. And the rest of the expression is plus eight minus ten x.
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And now to simplify this expression, we again will consider the Commutative Property and
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the Distributive Property. Now I’m not going to rewrite the expression this time, but we know that
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I could use the Commutative Property to rewrite it and put this minus ten x next to the minus twelve x.
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I'll just move into using the Distributive Property. So I have minus twelve and minus ten.
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So minus twelve minus ten times x, and minus eight plus eight is zero.
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Here, I have minus twelve minus ten, which is negative twenty-two, x.
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This is our simplified expression. In this next example, I have two sets of
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parentheses. So we will use the Distributive Property twice to remove the parentheses.
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Three times two p is six p, and three times negative nine is negative twenty-seven.
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Here, we have a minus sign in front of these parentheses which is just the same as multiplying
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or having a negative one in front of those parentheses. So I am distributing a negative one into the parentheses p plus ten.
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This will give us negative p minus ten. Now combining our like terms, we have a six p and a minus p that are like terms.
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And so using the Distributive Property, I can write that as six minus one
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times p. And I have a minus twenty-seven and a minus ten remaining. That combines
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to give us a negative thirty-seven, or minus thirty-seven. I have now six minus one which is
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five p. Six minus one which is five times p minus thirty-seven. This is our simplified expression.
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In our last example, we have one set of grouping symbols. We have the minus sign in front of the
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grouping symbol, so there is an implied one, or minus one, in front of those parentheses.
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So I will distribute a negative one into these parentheses, giving us a negative five x y,
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negative one times two z is negative two z, and I still have a plus three z
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minus nine x y. Let's look for our like terms. I'm looking for the same variables
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raised to the same powers. So I have an x y here and an x y here. These are like terms.
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Using the Distributive Property with those two terms, I have minus five minus nine times x y.
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And then I also have like terms here, z to the first power and z to the first power.
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So using the Distributive Property, I have plus a negative two plus three z.
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Now I have minus five minus nine, which is negative fourteen, then times x y.
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And negative two plus three is one, so this will be one z, or just z. This is our final, simplified expression.
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In these next examples, we are going to write the sentence as an algebraic expression,
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and then simplify. In our first example, we are going to add four x plus two to six
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x plus five. So we're beginning with six x plus five and adding four x plus two.
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Now because addition is commutative, the order in which I added these expressions didn't matter. I
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could have begun with a four x plus two or written this as four x plus two plus six x plus five.
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Now to simplify. Because the Associative Property and the Commutative Property both apply to
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addition, I can drop the parentheses. And now I can apply the Distributive Property. Combining my
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like terms six x plus four x, this would give me six plus four x, and then I have five plus two,
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which we'll add in our next step. Here, I have six plus four, which is ten, times x
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plus seven. So our final simplified expression is ten x plus seven.
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In our next example, we are to subtract four x plus two from six x plus five. So it's important
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to note here that we are beginning with six x plus five, and we are subtracting four x plus
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two from it. Now here the order is very important because subtraction is not commutative. It is only
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commutative if I were to rewrite a subtraction problem as addition of the opposite sign.
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Now I'm going to clear the parentheses by utilizing the Distributive Property.
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In this first set of parentheses, I'm multiplying by one. So there is nothing to do other than to
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rewrite the expression six x plus five. With this second set of parentheses, there is an implied
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one in front of the parentheses. And so we are going to distribute a negative one
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into the parentheses, giving us negative four x minus two. And now using the Distributive
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Property, I can rewrite this problem as six minus four times x plus five minus two.
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This gives me two x plus three, and this is our final, simplified expression.
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When presented with a problem, it is often useful to translate phrases into algebraic expressions.
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So let's take a look at these first two examples. The directions are to write the phrase as an
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algebraic expression. In this first example, we have one-third of a number decreased by two. So
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we'll begin with one-third of a number. Let's call that number x. This would be one-third x, and
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we are decreasing that one-third of a number by two. Decreasing by two means that we are subtracting
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two. One-third x minus two is our algebraic expression that is represented by this phrase.
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In the next example, we have nine times the sum of a number and three.
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So we're going to start with the sum of a number and three. Again, let's let that number be
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represented by x. So the sum of a number and three is x plus three. We need nine times that sum,
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so that would look like nine times x plus three in parentheses. Nine times the sum of x plus three.
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Let's look at this example. A plot of land is in the shape of a rectangle.
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If one side is four x minus two meters and one side is five x minus six meters, express the
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perimeter of the lot as a simplified expression in x. Let's draw a sketch to represent this problem.
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The shape is a rectangle, and one of the sides is four x minus two. I don't know whether that's
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the short side or the long side, but for the purpose of the sketch it really doesn't matter.
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So I will put this side as four x minus two. And one side is five x minus six.
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Well, I know that in a rectangle opposite sides have the same length,
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so this side here has to still be four x minus two. So the five x minus six I will put here.
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Now to write the perimeter, I know the perimeter means the length around,
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which perimeter of a rectangle can be represented as two times the length plus two times the width.
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Or it could be the sum of each of the four sides. So if I want to write an expression for the perimeter, I'm going to
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calculate two L plus two W. That would be two times four x minus two plus two times five x minus six. This expression
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represents the perimeter. Now let's simplify it. We're going to use the Distributive Property
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first to clear the parentheses. This will give us eight x minus four plus ten x minus twelve.
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And now combining like terms, again we'll use the Distributive Property, combining eight plus ten x.
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And I have minus four minus twelve. And so this would give us eighteen x, and minus four minus twelve, minus sixteen.
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So an expression for our perimeter is eighteen x minus sixteen meters.