 Polynomials are the analogues of the whole numbers, and what that means is that all the arithmetic you can do with the whole numbers corresponds to algebra you can do with polynomials. We can add, subtract, multiply, and now let's divide. To begin with, the division of polynomial expressions relies on the following ideas from the arithmetic of fractions. For b not equal to zero, a divided by b is the same as the fraction a over b, and for cd not equal to zero, a b over cd is the same as a over c times b over d. So let's consider 15x to the third divided by 3x squared. Now the first thing to recognize here is that there's a number of multiplications and divisions in this statement, so let's break those apart. 15x cubed is the same as 15 times x cubed, and 3x squared is the same as 3 times x squared. So now let's consider our order of operations. Exponents are supposed to be taken care of first, so we have to evaluate x to the third, which is x to the third, and likewise x to the second is just x to the second. We still have the operations 15 times x to the third and 3 times x to the second, but here it's important to remember multiplication and division are performed at the same time and from left to right, so we have to perform this multiplication 15 times x to the third first. So just as a reminder that we're supposed to do this first, let's throw that inside a set of parentheses, then multiply. Arithmetic is bookkeeping algebra generalized arithmetic, so everything else that we had before is still there. We still have divide by 3 times x squared. And again, multiplication and division are performed at the same time from left to right. We encounter this divide by 3, so we have to take care of the 15x cubed divided by 3 first. And again, to emphasize that, we'll throw it inside a set of parentheses. So 15x to the third divided by 3, well, a quotient can be rewritten as a fraction. So 15x cubed divided by 3 can be rewritten as 15x cubed over 3. And as long as our numerator and denominator are factors, we can split the factors into separate fractions. So we'll take the numbers as one fraction, 15 over 3, and well, the only thing left is the x cubed. We had it before, we still have it, and we can simplify. This fraction 15 over 3, well, that's the same as 15 divided by 3, which is 5, and we still have the x to the third. And so this 15x to the third divided by 3 is the same as 5x to the third, and we still have our x squared. And finally, we can multiply x to the third times x to the second, because remember in exponential expressions, if the base is the same, we can add the exponents. So we can simplify this last product to be 5x to the fifth. Now to see the difference that a set of parentheses makes, let's take a look at 15x to the third divided by parentheses 3x squared. So again, any quotient can be rewritten as a fraction. As long as the numerator and denominator are products, we can break the fraction apart into a product of fractions. So again, we have 15 times and 3 times, so we can split off the 15 over 3. And what's left over is x to the third over x to the second. We can find 15 over 3 again. Since we have a quotient of exponential expressions, we can apply our quotient rule for exponents, and that gives us x to power 1. And while we could write x to the first power, remember that any number raised to the power 1 is going to just be the number itself. Or we could take something more complicated, 30x to the fifth y squared divided by 12x squared y to the seventh. So we can rewrite the division as a fraction. The numerator and denominator consist of terms that are multiplied together, so we can split the fraction apart and rewrite it as a product of fractions. And we can then try to simplify each part separately. So let's take a look at that first fraction, 30 over 12, and see what we can do to simplify it. So remember, if there's a common factor in numerator and denominator, we can remove it. So 30 is 6 times 5, and 12 is 6 times 2. So we can remove the common factor and reduce the fraction. And equals means replaceable, so instead of 30, 12s, I'll write 5 halves. This next fraction, x to the fifth over x to the second, I can use the quotient rule for exponents to simplify that. So that's going to be x to the power 5 minus 2, otherwise known as x to the third. The last fraction, y to the second over y to the seventh. So we can use the quotient rule for exponents to simplify. And while there's nothing really wrong with this expression, we do prefer to avoid negative exponents. So remember 1 over a to the k is the same thing as a to the power minus k. So we can rewrite this expression y to minus 5 as 1 over y to the fifth. And finally we can multiply all of our terms together. And it's useful to remember if I multiply a factor times a fraction, the factor becomes a numerator and the denominator becomes the denominator. So this product 5 halves times x to the third times 1 over y to the fifth becomes 5x to the third over 2y to the fifth.