Polynomial Division and Rational Expressions
As you'll recall from that time you became the grandmaster lord high poobah of all things fractions, multiplying fractions requires us to multiply the numerator by the numerator and the denominator by the denominator.
Simple stuff. After multiplying numerators and denominators, we simplify if there's any simplifying to be done. Even simpler.
Sometimes we can simplify each fraction before we multiply them.
To multiply , first simplify the individual fractions to find .
That will save us some serious pencil graphite in the long run. Think of how many dozens of pennies you could save over the course of your lifetime if you follow this rule. Seriously though, it will make your life easier, even if it doesn't make you a pencil baron. When the fractions involve big numbers, simplifying can be annoying, and you don't need that kind of aggravation. How irked would you be to need to deal with this baddie:
Thankfully, we don't need to bother finding 7 × 121, nor paying someone to find it for us. Instead, we factor the numbers first and multiply second:
Now we can see which factors cancel, plain as day:
This was almost fun. We did say "almost," though, so let's not get too crazy here. We were able to cross out a lot of stuff and got a nice answer without having to bother with multiplication. That's at least as entertaining as a game of Boggle.
We can do similar things with rational expressions. To multiply two rational expressions, we multiply the numerators and multiply the denominators. We'll see if we can have even more fun this time around. Ladies and gentlemen, don your party hats.
While we could multiply out the numerator and the denominator for quadratic polynomials, why bother? We've already spent enough time on our quads today. Besides, that's another step of arithmetic where mistakes can sneak in, assuming you're not perfect. If your teacher is okay with answers in this form, don't make extra work for yourself. Do the bare minimum, and continue to scrape by in life until someone notices.
Just kidding! We're kidding so hard.
Be careful: When asked to multiply two rational expressions, factor them first. This will let you quickly and easily see which factors cancel out. It will also give you an opportunity to wear your "Factor Fiction?" T-shirt.
Dividing rational expressions is exactly like multiplying them, except that we turn the second expression upside-down. Make sure the lid is sealed tightly before you attempt this. You don't want your variables going everywhere.
Remember how dividing fractions works? if we want to divide two fractions, we multiply the first fraction by the reciprocal, or multiplicative inverse, of the second fraction.
To find , we multiply by the reciprocal of :
We're sure it comes as a big surprise that this is how division with rational expressions works as well. We can even picture you making your "surprised" face.
As with multiplication, we factor expressions before we divide them so that we can see which factors cancel out.
First we factor:
Then we change division to multiplication by taking the reciprocal:
Finally, we cancel factors wherever possible:
One other tip that's useful for division of rational expressions is that any polynomial can be written as a rational expression by putting the polynomial over 1. Note: Polynomials under 1 are probably still on soft food or formula. That information is neither here nor there, but we thought that you should know.