Mastering College Algebra: Simplifying Expressions with Ease

The world of algebra can sometimes feel like a maze of numbers and letters, can’t it? If you’re gearing up for the College Algebra CLEP exam, understanding how to simplify expressions is crucial. Let’s explore one such expression together, unraveling its complexities and making sense of it step-by-step.

We’re diving into the simplification of the expression ((x^3 + 7x^2 - 4x + 2)/(x - 2)). Is it starting to look daunting? Fear not! With a little practice in long division, you’ll be right on track like a pro. First off, let's recall that long division in algebra works similarly to the long division you learned back in elementary school—except now, it’s in the language of variables.

Step 1: Let’s Break it Down

Start by dividing the first term of the numerator, which is (x^3), by (x) (the first term of the denominator). The result? (x^2). Now, isn’t that satisfying? You’ve already made a significant step!

Step 2: Multiply and Subtract

Next, multiply (x^2) by ((x - 2)), yielding (x^3 - 2x^2). Take a moment to appreciate that—seeing how things fit together is a joy in itself! Now, you subtract this from the original expression:

[ (x^3 + 7x^2 - 4x + 2) - (x^3 - 2x^2) = 9x^2 - 4x + 2 ]

Have you ever seen how everything seems to work out? It’s like putting together pieces of a puzzle. Next, we focus on (9x^2).

Step 3: Keep Going!

Divide (9x^2) by (x) to get (9x). Multiply this result by ((x - 2)) to get (9x^2 - 18x). Time for another round of subtraction:

[ (9x^2 - 4x + 2) - (9x^2 - 18x) = 14x + 2 ]

Here we go again! We're not done yet, though. Now take (14x) and divide it by (x) to get (14). How cool is that?

Step 4: Final Touches

Multiply (14) by ((x - 2)) to give you (14x - 28). Time to subtract once more:

[ (14x + 2) - (14x - 28) = 30 ]

What do we have at the end of this journey? We’ve simplified our expression to (x^2 + 9x - 2).

In a Nutshell:

One could say, the process of long division in algebra is like a dance: you learn the steps, you fall out of rhythm, but you pick it back up with practice. Algebra doesn’t have to be an intimidating giant; with patience and the right tools, it can transform into a partner that guides you to success.

So, next time you're bent over your algebra homework, just remember this: break it down step-by-step, embrace the rhythm, and trust in your ability to find clarity. You've got this! If nothing else, just keep this expression in mind as an example to imitate, and soon enough, you won’t just be simplifying expressions; you’ll be dancing through math like it’s a piece of cake!