Simplify (x+10)/(x^2+7x-18) * (3x^2-12x+12)/(3x+30)

by Kenji Nakamura 52 views

Hey there, math enthusiasts! Today, we're diving into the fascinating world of algebraic expressions, specifically focusing on simplifying a product of rational expressions. The expression we're tackling is:

(x+10)/(x^2+7x-18) * (3x^2-12x+12)/(3x+30)

At first glance, it might seem a bit intimidating, but don't worry, we'll break it down step by step, making sure everyone understands the process. Our ultimate goal is to find the simplest form of this expression, which means we want to reduce it to its most basic form, where no further simplification is possible. So, let's put on our mathematical hats and get started!

1. Factoring: The Key to Simplification

The cornerstone of simplifying rational expressions lies in factoring. Factoring allows us to break down complex polynomials into simpler products, which then allows us to identify common factors that can be canceled out. Think of it like reducing a fraction – you find common factors in the numerator and denominator and divide them out. We're going to do the same thing here, but with polynomials!

1.1 Factoring the Denominator: x^2 + 7x - 18

Let's start with the denominator of the first fraction: x^2 + 7x - 18. This is a quadratic expression, and we need to find two numbers that multiply to -18 and add up to 7. After a little thought, we can identify those numbers as 9 and -2. Therefore, we can factor the quadratic as:

x^2 + 7x - 18 = (x + 9)(x - 2)

1.2 Factoring the Numerator: 3x^2 - 12x + 12

Next up is the numerator of the second fraction: 3x^2 - 12x + 12. The first thing we should always look for when factoring is a common factor among all the terms. In this case, we see that each term is divisible by 3. Factoring out the 3, we get:

3x^2 - 12x + 12 = 3(x^2 - 4x + 4)

Now we have another quadratic expression inside the parentheses: x^2 - 4x + 4. This one is a perfect square trinomial! It can be factored as:

x^2 - 4x + 4 = (x - 2)(x - 2) = (x - 2)^2

Putting it all together, we have:

3x^2 - 12x + 12 = 3(x - 2)^2

1.3 Factoring the Denominator: 3x + 30

Finally, let's factor the denominator of the second fraction: 3x + 30. Again, we look for a common factor, which is 3. Factoring out the 3, we get:

3x + 30 = 3(x + 10)

1.4 Factoring the Numerator: x + 10

The numerator of the first fraction is x + 10. This is already in its simplest form and cannot be factored further. Sometimes, you'll have expressions that are already factored, so it's a quick win!

2. Rewriting the Expression with Factored Forms

Now that we've factored all the components, let's rewrite the original expression using the factored forms:

(x+10)/(x^2+7x-18) * (3x^2-12x+12)/(3x+30) = (x + 10)/((x + 9)(x - 2)) * (3(x - 2)^2)/(3(x + 10))

This looks much more manageable already, doesn't it? We've transformed the expression from a seemingly complex one into a product of factored terms.

3. Canceling Common Factors: The Art of Reduction

This is where the magic happens! Now we can cancel out common factors that appear in both the numerator and the denominator. This is the essence of simplifying rational expressions.

3.1 Identifying and Canceling

Looking at our expression:

(x + 10)/((x + 9)(x - 2)) * (3(x - 2)^2)/(3(x + 10))

We can see several common factors:

  • (x + 10) appears in both the numerator and the denominator.
  • 3 appears in both the numerator and the denominator.
  • (x - 2) appears in both the numerator (as (x-2)^2) and the denominator.

Let's cancel these common factors out. Remember, canceling a factor means dividing both the numerator and the denominator by that factor.

(x + 10)/((x + 9)(x - 2)) * (3(x - 2)^2)/(3(x + 10)) = (1)/((x + 9)(1)) * ((x - 2))/(1)

After canceling, we are left with:

(1)/((x + 9)) * ((x - 2))/(1)

3.2 Multiplying the Remaining Terms

Now, let's multiply the remaining terms together:

(1)/((x + 9)) * ((x - 2))/(1) = (x - 2)/(x + 9)

4. The Simplest Form: (x - 2) / (x + 9)

And there you have it! The simplest form of the given expression is:

(x - 2) / (x + 9)

This expression cannot be simplified further because there are no more common factors between the numerator and the denominator. We've successfully navigated the world of rational expressions and arrived at our simplified answer!

5. Restrictions and Excluded Values: A Word of Caution

Before we celebrate our victory, there's one more crucial aspect to consider: restrictions or excluded values. In the world of rational expressions, we need to be mindful of values that would make the denominator zero, as division by zero is undefined. These values are called excluded values or restrictions.

5.1 Identifying Excluded Values

To find the excluded values, we need to look back at the factored denominators in our original expression:

(x + 9)(x - 2) and 3(x + 10)

We need to find the values of x that would make these expressions equal to zero.

  • x + 9 = 0 => x = -9
  • x - 2 = 0 => x = 2
  • x + 10 = 0 => x = -10

Therefore, the excluded values are x = -9, x = 2, and x = -10. These are the values that would make the original expression undefined.

5.2 Stating the Restrictions

When we present our simplified expression, it's important to state these restrictions. We can write our final answer as:

(x - 2) / (x + 9), where x ≠ -9, x ≠ 2, and x ≠ -10

This tells us that the simplified expression is equivalent to the original expression for all values of x except -9, 2, and -10.

6. Why is Simplifying Important?

You might be wondering, why go through all this trouble to simplify an expression? Well, there are several compelling reasons:

6.1 Clarity and Understanding

A simplified expression is much easier to understand and work with. It reveals the underlying structure of the expression and makes it easier to analyze its behavior.

6.2 Further Calculations

Simplified expressions are essential for further calculations, such as solving equations, graphing functions, and performing calculus operations. Trying to work with a complex expression can lead to errors and make the process much more difficult.

6.3 Problem Solving

In many mathematical problems, simplifying expressions is a crucial step in finding the solution. It allows you to see patterns and relationships that might be hidden in the original form.

7. Practice Makes Perfect: Examples and Exercises

The best way to master simplifying rational expressions is through practice. Let's look at a couple of additional examples and then suggest some exercises for you to try on your own.

7.1 Example 1

Simplify the expression:

(2x^2 + 6x) / (x^2 + 5x + 6)

Solution:

  1. Factor the numerator: 2x^2 + 6x = 2x(x + 3)
  2. Factor the denominator: x^2 + 5x + 6 = (x + 2)(x + 3)
  3. Rewrite the expression: (2x(x + 3)) / ((x + 2)(x + 3))
  4. Cancel common factors: (2x(x + 3)) / ((x + 2)(x + 3)) = (2x) / (x + 2)
  5. State the restrictions: x ≠ -2, x ≠ -3
  6. Simplified expression: (2x) / (x + 2), where x ≠ -2, x ≠ -3

7.2 Example 2

Simplify the expression:

(x^2 - 4) / (x^2 - 4x + 4)

Solution:

  1. Factor the numerator: x^2 - 4 = (x + 2)(x - 2) (Difference of Squares)
  2. Factor the denominator: x^2 - 4x + 4 = (x - 2)^2
  3. Rewrite the expression: ((x + 2)(x - 2)) / ((x - 2)^2)
  4. Cancel common factors: ((x + 2)(x - 2)) / ((x - 2)^2) = (x + 2) / (x - 2)
  5. State the restrictions: x ≠ 2
  6. Simplified expression: (x + 2) / (x - 2), where x ≠ 2

8. Conclusion: Mastering the Art of Simplification

Simplifying rational expressions is a fundamental skill in algebra, and it's one that you'll use again and again in your mathematical journey. By mastering the art of factoring, identifying common factors, and stating restrictions, you'll be well-equipped to tackle a wide range of algebraic problems.

Remember, the key is practice! Work through examples, try exercises, and don't be afraid to ask for help when you need it. With consistent effort, you'll become a simplification pro in no time!

So, guys, keep practicing, keep exploring, and keep simplifying those expressions! You've got this!