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Adding and subtracting polynomials of degree two | Algebra 1 (TX TEKS) | Khan Academy

2m read
·Nov 10, 2024

So we have two different expressions here, and what I want you to do is pause this video and see if you can rewrite each of these as a simplified polynomial in standard form. So pause the video and have a go with that.

All right, now let's do this together. So this first one we are adding two polynomials, and I could just rewrite this as -5x + 4x² + 7. Since I'm adding this entire second polynomial, I could just say this is going to be + 3x - 6 - 8x².

Now the key is we want to combine like terms. What do I mean by like terms? The ones that are the same degree. For example, here I have 4x², and then I am subtracting 8x². So if I have four of something and I subtract eight of that same something, I am now going to have -4 of that something. In this case, that something is x².

Now let's go to the first degree terms. I have -5x's and I also have 3x's. So if I take 3x's and I subtract 5x's, well, I'm going to have -2x's. And then last but not least, I have our constant terms. If I have a 7 and I subtract 6 from that, I am going to be left with 1.

And there I have it; I've simplified it. It's a polynomial, and it's in standard form. I've put the highest degree term first, the second degree term, then the first degree term, and then the constant term.

Let's do the same thing with this one. Now this one I can rewrite this first polynomial, the first part of this expression, as 5y + 3y² - 9. But we have to be a little bit careful here because here we are subtracting this second polynomial.

Another way to think about it is we could view this as if I'm subtracting it; that's the same thing as 1 times all of this. So if I want to remove these parentheses, I have to distribute this -1 onto every term.

So, -1 * 8y² is -8y², -1 * -1 is +1, -1 * 2y is -2y. Now I can do what I just did in the previous example. I could, for example, say, all right, where are my second degree terms? I have 3y², and I'm going to subtract 8y² from that. Well, that's going to be -5y².

Then I could go to our first-degree terms. I have 5y's, and then from that, I'm going to subtract 2y. Well, that's going to give me +3y. And then last but not least, I have -9 here, and then I'm going to add 1, which would get us to -8.

And we're done.

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