 So what happens when we divide a polynomial by a monomial? So remember, a polynomial is a sum of monomial terms. And so we might consider what happens when we have a fraction where the numerator is a sum. And so this goes back to a theorem for c not equal to 0. The fraction a plus b over c is a over c plus b over c. Now we're used to reading this from right to left, because this is how we add fractions. They have to have the same denominator. But we can also use it backwards to split up a fraction whose numerator is a sum. So let's divide 15x plus 3 by 3x. So again, the order of operations requires that multiplication and division be done before addition and subtraction. So if we read this, we have 15x plus 3 divided by 3x. We have to do the 15 times x first, done. And then the 3 divided by 3x next. To emphasize this point, we'll throw that division inside its own set of parentheses. So 3 divided by 3x, well, that's the same as the fraction 3 over 3x. So we can rewrite our expression. And we can simplify that. A useful thing to remember here is that any number a can be rewritten as a times 1. So I can rewrite my numerator 3 as 3 times 1. My denominator is 3 times x. And now I can rewrite this fraction as a product. 3 over 3 times 1 over x. Now I can simplify. 3 over 3 is 1. And so we don't really need to include a factor of 1. And 1 over x is 1 over x. Equals means replaceable. So instead of 3 over 3x, I can write 1 over x. And that's as far as we can go in simplifying this expression. Again, parentheses make a big difference. Here, I'm taking 15x plus 3. And all of that is being divided by 3x. So I can rewrite this as a fraction. Since my numerator is a sum, I can split this into two fractions, 15x over 3x plus 3 over 3x. 15x over 3x. Since numerator and denominator are both products, we can split this into a product of two fractions. So let's rewrite it and simplify. And from the previous problem, we already figured out what 3 over 3x is. And so our final expression, 5 plus 1 over x. Well, let's take a look at another division. So again, every division can be rewritten as a fraction. Since this is a subtraction, we can split our fraction into two parts with the same denominator. And then we can simplify each part. 8x to the 12th over 2x to the fourth. Because numerator and denominator are products, we can split this up into a product of fractions. And then simplify. Similarly, 4x squared over 2x to the fourth, numerator and denominator are both products of terms. So we can rewrite this as a product of fractions. And simplify. And again, we'd prefer not to have negative exponents, so we'll take it one further step. I lie, we'll take it one more step. And so instead of 8x to the 12th over 2x to the fourth, we can write 4x to the eighth. And instead of 4x to the second over 2x to the fourth, we can write 2 over x squared. And we can do this with more terms. We'll rewrite our division as a fraction. Since we're adding and subtracting in the numerator, we can rewrite this as an addition or subtraction of fractions with the same denominator. And now each of our fractions is just a product over a product, so we can simplify by rewriting each fraction as a product. We'll rewrite our first rational expression as a product, separating the variables. Our first fraction, 12 over 2, can be reduced. x squared over x can be reduced. We're kind of stuck with the 1 over y, so we'll leave that alone, but we'll reduce z to the fourth over z. And we'll rewrite our product as a single rational expression. And if it's not written down, it didn't happen. We'll rewrite our second rational expression as a product. We'll simplify the fraction for over 2. x over x, y over y, and z over z are all equal to 1, so this reduces to just 2. For the third rational expression, we'll rewrite it as a product. Then simplify and combine them back into a single rational expression.