Question Video: Simplifying an Algebraic Expression by Multiplying Monomials by Binomials Mathematics

Simplify the expression 2π‘₯(π‘₯𝑦 + 𝑦) βˆ’ 𝑦(2π‘₯ βˆ’ 𝑦) βˆ’ π‘₯Β²(2𝑦 βˆ’ 1).

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Video Transcript

Simplify the expression two π‘₯ times π‘₯𝑦 plus 𝑦 minus 𝑦 times two π‘₯ minus 𝑦 minus π‘₯ squared times two 𝑦 minus one.

First, we copy down our expression. In this expression, everywhere we see parentheses, we’ll have to do some distribution. So let’s start here. We need to distribute this multiplication across both terms in the parentheses. That means we’re multiplying two π‘₯ times π‘₯𝑦. And then we’ll be adding two π‘₯ times 𝑦. Now, for our second set of parentheses, we’ll need to be a bit more careful because we’re multiplying by negative 𝑦. And that means we’ll have negative 𝑦 times two π‘₯.

We want to write this as plus negative 𝑦 times two π‘₯. And then we’ll multiply negative 𝑦 by negative 𝑦. So we’ll write that as plus negative 𝑦 times negative 𝑦. And when we come to the third set, we see the same thing. We have a negative π‘₯ squared we’re multiplying. And we’ll write that as plus negative π‘₯ squared times two 𝑦 and then negative π‘₯ squared times negative one, which we write as plus negative π‘₯ squared times negative one.

This step that I’ve put here is kind of an intermediate step. Once you get really good at these type of problems, you won’t have to write this out all the way. Instead, you would just say two π‘₯ times π‘₯𝑦 equals two π‘₯ squared 𝑦 and two π‘₯ times 𝑦 is two π‘₯𝑦. Negative 𝑦 times two π‘₯ we would rewrite as negative two π‘₯𝑦. This is because we generally list the coefficient first. And the order of π‘₯ and 𝑦 doesn’t have to be like this, but it’s good to keep that consistent throughout the same expression. From there, negative 𝑦 times negative 𝑦 is positive 𝑦 squared. And negative π‘₯ squared times two 𝑦 we would write as negative two π‘₯ squared 𝑦. And finally, negative π‘₯ squared times negative one we write as positive π‘₯ squared.

We’ve done all of our expanding and multiplying, but our instructions tell us to simplify. When we look at the term two π‘₯ squared 𝑦, we also see another like term negative two π‘₯ squared 𝑦. And when we add these two values together they equal zero. We also recognize we have the term two π‘₯𝑦 and negative two π‘₯𝑦. When we add those together, we get zero, which means our remaining values are 𝑦 squared and π‘₯ squared. So we could write the simplified form of this expression as π‘₯ squared plus 𝑦 squared.

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