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Simplifying Expressions


Our first topic in algebraic expressions is simplifying expressions. So suppose we were asked to simplify 3x + 5x. Now, of course, this is a very easy question. The test is not gonna ask something this basic, but suppose we have to simply 3x + 5x. Of course, 3x + 5x = 8x.

Now, let's think about this for a minute. We've said one expression, 3x + 5x, is equal to another expression, 8x, so what we're saying here is we have a rule that's true for all numbers. Every single number on the number line would obey this particular rule. And of course, we can add 3x + 5x using the Distributive Law because they are like terms.

Now, what do we mean by like terms? Like terms are any two terms with the same variable part. If they differ at all, they differ only in coefficients. The number one algebraic simplification rule, we can combine like terms by addition or subtraction. And in essence, we're just adding or subtracting the coefficients.

So for example, 15y- 8y. Both of those are like terms. The variable part is just y, so we can subtract. 15- 8 is 7, so that's 7y. 3y squared plus 3y squared, those are like terms, in fact they're identical. So when we add, we just get 3 plus 3 which is 6, 6y squared.

In that final one, notice that thing is awfully complicated. But the two terms are actually like terms, because the variable part, x to the 5th, y to the 4th, z to the 7th, those are identical between the two terms. So we have two of the thing, plus one of the thing, and of course whenever you have two of anything plus one of the same thing, you get three of that thing. So that's how we simplify that.

Notice that we can add or subtract only like terms, not terms with different variables or powers. So if we look at this expression, we can't just add everything together, because what we have here are different powers of y. So we have to group the terms by the like term, by the powers of y. So the cubic, y cubed, that's gonna be by itself.

We group the two quadratic terms and the two linear terms. And then within those parentheses, within those groupings, we can simplify, so we get down to this simplified expression. Also, notice that multiplication is commutative. In other words, a times b equals b times a. You can switch the order around and it doesn't change the multiplication.

So the order of factors in multiplication doesn't matter. Therefore, like terms may appear different if the multiplied variables are in a different order. So they're not really different, it's just a difference in appearance. So for example, 5xy + 7yx. Those are still like terms.

Because xy and yx are the same thing. The order doesn't matter, so those are like terms and we can add them. And even this one, the second one, this is more complicated, but notice that all we have are the same factors, the same variables just multiplied in a different order. So we have 6 of something minus 4 of the same thing, which would be 2 of that thing.

So now, pause the video and simplify these and then we'll talk about these. In the first one, we have to group the like terms and it simplifies to this. In the second one, we have to group the like terms and it simplifies to this. In the third one, it turns out there's no simplification possible, because no two terms are like terms. So, that's as simple as it gets right there.

Another simplification topic is how to combine algebraic expressions involving parentheses. Parentheses, this is a big mathematical topic, parentheses. When an addition sign appears in front of the parentheses, we can simply remove the parentheses. That's very easy, so we have this, then we can just write the exact expression again as if there were no parentheses at all, just basically erase the parentheses, and then we can do our simplifying from there.

And it turns out this simplifies just to the single cubic monomial 2x cubed. When a subtraction sign appears in front of the parentheses, things are a little trickier. In removing the parentheses, we have to change every sign inside the parentheses to its opposite. Addition inside the parentheses becomes subtraction outside, and vice versa.

So here, we're subtracting those parentheses, so the first parentheses we could just remove without any problem, but the second parentheses, when we remove those, the addition of 3x squared inside the parentheses becomes subtraction outside. And the subtraction, the subtraction of 3x becomes addition outside. Those two change to its opposite.

And once we have that, then we can simplify and it simplifies to this. Pause the video here and simplify these expressions. And these are the answers. So, in summary, we can simplify algebraic expressions by adding, subtracting like terms, that's a really big idea. When adding an expression in parentheses, we can simply remove the parentheses.

When subtracting an expression in parentheses, we have to change each term to its opposite sign when we remove the parentheses.

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