Simplify Rational Expression & Excluded Values: Step-by-Step

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Hey guys! Let's break down this rational expression problem step by step. We've got the expression (x² - 2x - 15) / (x² - 9), and our mission is twofold: First, we need to find an equivalent expression in its simplest form. Second, we've got to figure out which values of 'x' would make this expression go boom (aka, be undefined) and exclude them from the domain. So, let's dive in and make math a little less mysterious!

1. Finding the Equivalent Expression in Lowest Terms

To get this rational expression into its simplest form, we need to factor both the numerator and the denominator. Factoring is like reverse-engineering multiplication; we're trying to find the expressions that, when multiplied together, give us the original quadratic expressions. This is a crucial step, so let's take our time and get it right. Factoring helps us identify common factors that we can cancel out, simplifying the expression.

Factoring the Numerator (x² - 2x - 15)

Okay, let's tackle the numerator first. We need to find two numbers that multiply to -15 and add up to -2. Think of the factors of 15: 1 and 15, 3 and 5. Which pair can give us -2 when combined with the right signs? Bingo! 3 and -5. So, we can rewrite the numerator as:

(x - 5)(x + 3)

This means if we were to multiply (x - 5) and (x + 3), we'd get back our original numerator, x² - 2x - 15. Factoring is like unlocking a puzzle, and this is the first piece we've put into place.

Factoring the Denominator (x² - 9)

The denominator, x² - 9, should ring a bell for those familiar with difference of squares. It's in the form a² - b², which factors neatly into (a - b)(a + b). In our case, a is x and b is 3 (since 9 is 3²). So, the denominator factors into:

(x - 3)(x + 3)

See how elegant that is? Recognizing these patterns makes factoring so much smoother. Now we've got both the numerator and the denominator in factored form, and we're ready for the simplification magic.

Simplifying the Expression

Now that we've factored both the numerator and the denominator, our expression looks like this:

(x - 5)(x + 3) / (x - 3)(x + 3)

Here's where the simplification happens. Notice that we have a common factor of (x + 3) in both the numerator and the denominator. We can cancel these out, just like simplifying a fraction like 6/8 by dividing both by 2. So, we wave goodbye to (x + 3), and we're left with:

(x - 5) / (x - 3)

Ta-da! This is our equivalent expression in lowest terms. We've successfully simplified the original rational expression by factoring and canceling common factors. But our mission isn't over yet. We still need to identify those pesky values of x that we need to exclude.

2. Identifying Excluded Values from the Domain

The domain of an expression is all the possible values of x that we can plug in without causing any mathematical mayhem. In the case of rational expressions, the main thing we need to watch out for is division by zero. Dividing by zero is a big no-no in the math world; it makes the expression undefined.

So, to find the excluded values, we need to look at the original denominator (before we simplified) and figure out what values of x would make it equal to zero. Why the original denominator? Because even though we simplified the expression, the excluded values are still determined by what would make the original expression undefined.

Setting the Original Denominator to Zero

Our original denominator was x² - 9. We need to solve the equation:

x² - 9 = 0

We already factored this as (x - 3)(x + 3), so we can rewrite the equation as:

(x - 3)(x + 3) = 0

To solve this, we use the zero product property, which says that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero:

x - 3 = 0 or x + 3 = 0

Solving these simple equations gives us:

x = 3 or x = -3

These are our excluded values! If we plug x = 3 or x = -3 into the original expression, the denominator becomes zero, and the expression is undefined. We've found the values of x that we need to keep out of the domain to avoid mathematical chaos.

3. The Answers and Why

So, to recap, we've simplified the rational expression and identified the excluded values. Let's put it all together:

  1. The equivalent expression in lowest terms is (x - 5) / (x - 3).
  2. The values of x that must be excluded from the domain are x = 3 and x = -3.

Therefore, the correct answers from the choices provided are:

  • A. x = -3
  • C. x = 3

We exclude these values because they make the denominator of the original expression equal to zero, resulting in an undefined expression. This is a fundamental concept in dealing with rational expressions, and understanding it ensures we're not dividing by zero, which is a mathematical no-no!

Key Takeaways

Let's solidify our understanding with some key takeaways:

  • Factoring is Key: Simplifying rational expressions hinges on your ability to factor both the numerator and the denominator. Practice your factoring skills – it's a superpower in algebra!
  • Difference of Squares: Recognizing patterns like the difference of squares (a² - b²) can make factoring much faster and easier.
  • Excluded Values Come from the Original Denominator: Always look at the original expression when determining excluded values. Even if you simplify, the values that made the original denominator zero are still excluded.
  • Division by Zero is a No-Go: Remember, a rational expression is undefined when the denominator is zero. This is the golden rule for finding excluded values.

Level Up Your Understanding

Want to take your understanding of rational expressions to the next level? Here are a few ideas:

  • Practice, Practice, Practice: Work through more examples of simplifying rational expressions and finding excluded values. The more you practice, the more comfortable you'll become.
  • Explore Different Factoring Techniques: Brush up on different factoring methods, such as factoring by grouping and factoring trinomials with leading coefficients other than 1.
  • Graph Rational Functions: Visualizing rational functions can give you a deeper understanding of how excluded values affect the graph. Look for vertical asymptotes at the excluded values.
  • Real-World Applications: Think about how rational expressions might be used in real-world scenarios, such as calculating rates, proportions, or concentrations.

Conclusion

And there you have it! We've successfully simplified a rational expression and identified the values that need to be excluded from its domain. Remember, the key is to factor, simplify, and conquer! By understanding these concepts, you'll be well-equipped to tackle more complex algebraic problems. Keep practicing, stay curious, and happy math-ing, guys!