Simplifying Rational Expressions: A Step-by-Step Guide

Hey guys! Let's dive into simplifying a complex rational expression. It looks intimidating at first, but we'll break it down step by step. Our mission is to simplify this expression: $\frac{x^2+x-6}{x^2-7 x-30} \cdot \frac{(x+2)^2-4}{x^2-5 x+6} \cdot \frac{x^2-13 x+30}{x^3+4 x^2}$. Buckle up, it's gonna be a fun ride!

Step 1: Factor Everything

The first thing we need to do is factor every single polynomial we see. Factoring is like reverse engineering; we're trying to find out what two (or more) expressions multiply together to give us the original polynomial. This is the most crucial step, so take your time and double-check your work. Accuracy is key!

Factoring $x^2 + x - 6$

We need two numbers that multiply to -6 and add to 1. Those numbers are 3 and -2. So, we can factor this quadratic as:

$x^2 + x - 6 = (x + 3)(x - 2)$

Factoring $x^2 - 7x - 30$

Here, we need two numbers that multiply to -30 and add to -7. Those numbers are -10 and 3. Thus:

$x^2 - 7x - 30 = (x - 10)(x + 3)$

Factoring $(x+2)^2 - 4$

First, expand $(x+2)^2$ to get $x^2 + 4x + 4$. Then, subtract 4:

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

Now, factor out an $x$:

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

Factoring $x^2 - 5x + 6$

We need two numbers that multiply to 6 and add to -5. Those numbers are -2 and -3. So:

$x^2 - 5x + 6 = (x - 2)(x - 3)$

Factoring $x^2 - 13x + 30$

We need two numbers that multiply to 30 and add to -13. Those numbers are -10 and -3. Therefore:

$x^2 - 13x + 30 = (x - 10)(x - 3)$

Factoring $x^3 + 4x^2$

We can factor out $x^2$ from both terms:

$x^3 + 4x^2 = x^2(x + 4)$

Step 2: Rewrite the Expression with Factored Forms

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

$\frac{(x + 3)(x - 2)}{(x - 10)(x + 3)} \cdot \frac{x(x + 4)}{(x - 2)(x - 3)} \cdot \frac{(x - 10)(x - 3)}{x^2(x + 4)}$

Step 3: Cancel Common Factors

This is where the magic happens! We can cancel out any factors that appear in both the numerator and the denominator. Remember, we can only cancel factors that are multiplied, not added or subtracted. Let's start canceling:

• $(x + 3)$ in the first fraction
• $(x - 2)$ in the first and second fractions
• $(x - 10)$ in the first and third fractions
• $(x - 3)$ in the second and third fractions
• $(x + 4)$ in the second and third fractions
• $x$ in the second and third fractions (leaving $x$ in the denominator of the third fraction)

After canceling, we are left with:

$\frac{1}{1} \cdot \frac{1}{1} \cdot \frac{1}{x}$

Step 4: Simplify

Now, multiply the remaining terms together:

$\frac{1}{x}$

So, the simplified expression is $\frac{1}{x}$. Awesome, right?

Step 5: State Restrictions (Important!)

We need to identify any values of $x$ that would make the original expression undefined. This happens when any of the denominators are equal to zero. So, we look back at our factored denominators and solve for $x$:

• $x - 10 = 0 \Rightarrow x = 10$
• $x + 3 = 0 \Rightarrow x = -3$
• $x - 2 = 0 \Rightarrow x = 2$
• $x - 3 = 0 \Rightarrow x = 3$
• $x^2 = 0 \Rightarrow x = 0$
• $x + 4 = 0 \Rightarrow x = -4$

Therefore, the restrictions are $x \neq -4, -3, 0, 2, 3, 10$. These are the values that $x$ cannot be, otherwise, the original expression would be undefined.

Final Answer

The simplified expression is $\frac{1}{x}$, with the restrictions $x \neq -4, -3, 0, 2, 3, 10$.

Key Concepts Used

• Factoring Quadratics: Breaking down polynomials into their multiplicative factors. Understanding how to factor quickly and accurately is crucial. Look for patterns like difference of squares, perfect square trinomials, and grouping.
• Simplifying Rational Expressions: Canceling common factors in the numerator and denominator. This requires a solid understanding of factoring. Be careful to only cancel factors that are multiplied.
• Identifying Restrictions: Finding values of $x$ that make the denominator zero. These values must be excluded from the domain of the simplified expression. Always go back to the original expression to find these restrictions.

Common Mistakes to Avoid

• Incorrect Factoring: Double-check your factoring! A single mistake here will throw off the entire problem. Use the FOIL method (First, Outer, Inner, Last) to verify your factored expressions.
• Canceling Terms Instead of Factors: You can only cancel factors (things that are multiplied). Don't try to cancel individual terms that are added or subtracted.
• Forgetting Restrictions: Always, always, always state the restrictions. It's an essential part of the problem and shows a complete understanding.
• Not Factoring Completely: Ensure that all polynomials are factored completely before attempting to cancel. Look for opportunities to factor out a Greatest Common Factor (GCF) first.

Practice Problems

Try simplifying these expressions to solidify your understanding:

1. $\frac{x^2 - 4}{x^2 + 4x + 4}$
2. $\frac{2x^2 + 5x - 3}{x^2 - 9}$
3. $\frac{x^3 - 8}{x^2 + 2x + 4}$

Work through these problems, and don't hesitate to review the steps we covered. With practice, you'll become a pro at simplifying rational expressions!

Conclusion

Simplifying rational expressions might seem daunting at first, but by breaking it down into smaller, manageable steps, it becomes much easier. Remember to factor everything, cancel common factors, and state the restrictions. Keep practicing, and you'll master this skill in no time. You got this, guys! Happy simplifying! Remember that practice makes perfect, so keep at it! Good luck! And always remember to have fun while learning! The more you enjoy the process, the easier it will become.