High School: Algebra
Arithmetic with Polynomials and Rational Expressions HSA-APR.D.6
6. Rewrite simple rational expressions in different forms; write a(x)⁄b(x) in the form q(x) + r(x)⁄b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
This standard is like a Rube Goldberg machine. It looks far more complicated than it really is.
It all boils down to dividing polynomials and expressing the answer properly. Students should have already touched division and fractions at least a little, having been through the third grade and all. In a way, it's just a continuation of the Remainder Theorem, so we recommend covering that first.
Students should already know how to divide polynomials by factoring or long division. As with many divisions, they won't all be perfect and a remainder will be left over. Instead of just writing what the remainder is, we now expect students to actually do something with it.
Let's say we're dividing a(x) by b(x), and our answer is q(x) with remainder r(x). Just like the Remainder Theorem, if r(x) = 0, then b(x) is a factor of a(x). We know that already.
But what if b(x) doesn't divide a(x) with remainder 0? Well, just like simplifying 13⁄4 to 3 with a remainder of 1, or 3¼, we can write a(x)⁄b(x) as q(x) with remainder r(x), or r(x)⁄b(x). Just like a remainder of 1 divided by 4 means ¼, a remainder of r(x) divided by b(x) will give us r(x)⁄b(x).
All the talk about the degree of r(x) being less than the degree of b(x) just means that r(x) should be "smaller" than b(x). It wouldn't make sense to split 13⁄4 into 2 5⁄4 because we can still divide 5 by 4. It's the same idea, only polynomial-style.
We suggest relating these polynomial quotients to fractions of integers so that students don't feel overwhelmed. It's understandable for them to be confused when we throw seven different functions at them, but they'll be a lot more receptive when they're working with concepts they already know.
Students should also not be afraid of the big bad long division monster. Often, factoring is near impossible to figure out when remainders are involved and there are times when synthetic division just won't cut it. Students should give in and embrace long division and their lives will be better for it.
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