Solving Algebraic Expressions A Step-by-Step Guide

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Hey guys! Let's dive into the world of algebraic expressions. It might seem daunting at first, but trust me, with a bit of practice, you'll be solving these like a pro. We're going to break down two complex expressions step by step, making sure you understand every little detail. So, grab your pencils, and let's get started!

Part a: Combining Rational Expressions with Different Denominators

In this first part, we're tackling the expression: (a²-9)/(2a²+1) + (6a+1)/(a-3) + (6a-1)/(a+3). This looks like a beast, I know, but we’ll tame it together. The main idea here is to combine these fractions, which means we need a common denominator. Think of it like adding regular fractions – you can’t add halves and thirds directly; you need to convert them to sixths first. Same principle applies here, just with more complex terms.

Finding the Common Denominator

Our denominators are 2a²+1, (a-3), and (a+3). The common denominator will be the product of these, which gives us (2a²+1)(a-3)(a+3). Yes, it's a mouthful, but don't worry, we’ll handle it. This common denominator is what each fraction will eventually have.

Adjusting the Fractions

Now, we need to adjust each fraction so it has this common denominator. This means multiplying both the numerator and the denominator of each fraction by the factors it's currently missing.

  1. For the first fraction (a²-9)/(2a²+1), we need to multiply both the numerator and denominator by (a-3)(a+3). Remember that (a-3)(a+3) is a difference of squares, which simplifies to a² - 9. So, we're multiplying by (a² - 9).
  2. For the second fraction (6a+1)/(a-3), we need to multiply both the numerator and denominator by (2a²+1)(a+3). This ensures that the denominator matches our common denominator.
  3. For the third fraction (6a-1)/(a+3), we multiply both the numerator and denominator by (2a²+1)(a-3).

Rewriting the Expression

After adjusting each fraction, our expression looks like this:

[(a²-9)(a²-9)] / [(2a²+1)(a-3)(a+3)] + [(6a+1)(2a²+1)(a+3)] / [(2a²+1)(a-3)(a+3)] + [(6a-1)(2a²+1)(a-3)] / [(2a²+1)(a-3)(a+3)]

Okay, it looks super long, but we’ve achieved the crucial step of having a common denominator. This means we can now combine the numerators.

Combining the Numerators

Now, let's combine those numerators. This involves some expanding and simplifying. It's where the real algebraic fun begins! We’ll take it step by step:

  1. Expand (a²-9)(a²-9): This gives us a⁴ - 18a² + 81. Just use the FOIL (First, Outer, Inner, Last) method or the distributive property to multiply it out.
  2. Expand (6a+1)(2a²+1)(a+3): First, multiply (6a+1)(2a²+1), which results in 12a³ + 6a + 2a² + 1. Then, multiply this by (a+3). This is a bit lengthy, but take your time and be careful with each term. You'll get 12a⁴ + 36a³ + 2a³ + 6a² + a + 3 + 6a + 1, which simplifies to 12a⁴ + 38a³ + 6a² + 7a + 3.
  3. Expand (6a-1)(2a²+1)(a-3): Similarly, start by multiplying (6a-1)(2a²+1), which gives 12a³ - 2a² + 6a - 1. Multiply this by (a-3). You should get 12a⁴ - 36a³ - 2a³ + 6a² + 6a² - 18a - a + 3, which simplifies to 12a⁴ - 38a³ + 12a² - 19a + 3.

Putting it All Together

Now, let's add all those expanded numerators:

(a⁴ - 18a² + 81) + (12a⁴ + 38a³ + 6a² + 7a + 3) + (12a⁴ - 38a³ + 12a² - 19a + 3)

Combine like terms:

  • a⁴ terms: a⁴ + 12a⁴ + 12a⁴ = 25a⁴
  • a³ terms: 38a³ - 38a³ = 0 (They cancel each other out! Nice and clean.)
  • a² terms: -18a² + 6a² + 12a² = 0 (Another cancellation! We love to see it.)
  • a terms: 7a - 19a = -12a
  • Constants: 81 + 3 + 3 = 87

So, our combined numerator is 25a⁴ - 12a + 87.

The Simplified Expression

Finally, our expression looks like this:

(25a⁴ - 12a + 87) / [(2a²+1)(a-3)(a+3)]

And that, my friends, is the simplified form of the first expression. You might be tempted to try and factor further, but in this case, the numerator doesn't factor nicely, and neither does the denominator beyond what we've already done. So, we’re done with part a!

Part b: Simplifying Rational Expressions with Differences of Squares

Now, let's move on to part b: (5x + y)/(x - 5y) + (5x-y)/(x+5y) + (x² + y²)/(x²-25y²). This one also involves combining fractions, but it has a neat twist – a difference of squares, which makes our lives a bit easier.

Spotting the Difference of Squares

Notice that the denominator x²-25y² is a difference of squares. It can be factored into (x - 5y)(x + 5y). This is a crucial observation because it immediately tells us our common denominator.

Finding the Common Denominator

The common denominator for this expression is (x - 5y)(x + 5y). This makes the problem much simpler because we already have it in the third fraction. We just need to adjust the first two.

Adjusting the Fractions

  1. For the first fraction (5x + y)/(x - 5y), we need to multiply both the numerator and the denominator by (x + 5y).
  2. For the second fraction (5x-y)/(x+5y), we need to multiply both the numerator and the denominator by (x - 5y).
  3. The third fraction (x² + y²)/(x²-25y²) already has the common denominator, so we can leave it as is.

Rewriting the Expression

After adjusting, our expression looks like this:

[(5x + y)(x + 5y)] / [(x - 5y)(x + 5y)] + [(5x-y)(x - 5y)] / [(x - 5y)(x + 5y)] + (x² + y²) / [(x - 5y)(x + 5y)]

Now we have a common denominator, so we can combine those numerators!

Combining the Numerators

Let’s combine and simplify the numerators:

  1. Expand (5x + y)(x + 5y): Using FOIL, we get 5x² + 25xy + xy + 5y², which simplifies to 5x² + 26xy + 5y².
  2. Expand (5x - y)(x - 5y): Using FOIL again, we get 5x² - 25xy - xy + 5y², which simplifies to 5x² - 26xy + 5y².

Putting it All Together

Now, let's add all the numerators:

(5x² + 26xy + 5y²) + (5x² - 26xy + 5y²) + (x² + y²)

Combine like terms:

  • x² terms: 5x² + 5x² + x² = 11x²
  • xy terms: 26xy - 26xy = 0 (They cancel out! Another win!)
  • y² terms: 5y² + 5y² + y² = 11y²

So, our combined numerator is 11x² + 11y².

The Simplified Expression

Our expression now looks like this:

(11x² + 11y²) / [(x - 5y)(x + 5y)]

Factoring and Simplifying Further

We can factor out an 11 from the numerator: 11(x² + y²). Our expression becomes:

11(x² + y²) / [(x - 5y)(x + 5y)]

And that’s as simplified as it gets! We can’t cancel anything else, so we’re done with part b!

Final Thoughts

And there you have it, folks! We've tackled two pretty complex algebraic expressions, breaking them down step by step. Remember, the key to mastering these problems is practice, patience, and a good understanding of the basic rules of algebra. Keep at it, and you'll be an algebraic wizard in no time!

If you ever get stuck, just remember to:

  • Find the common denominator.
  • Adjust the fractions.
  • Combine the numerators.
  • Simplify as much as possible.

Happy solving, and I'll catch you in the next one! Keep those algebraic skills sharp!