Simplify (150c^2 - 500c) / (3c - 10): A Step-by-Step Guide

by Alex Johnson 59 views

Let's dive into simplifying the algebraic expression (150c^2 - 500c) / (3c - 10). This kind of problem often appears in algebra and is a great way to practice your factoring and simplification skills. We will break it down into manageable steps, ensuring you understand each part of the process.

Understanding the Expression

The expression we're working with is a rational expression, which simply means it's a fraction where the numerator and the denominator are polynomials. In our case, the numerator is 150c^2 - 500c, and the denominator is 3c - 10. The goal is to simplify this fraction as much as possible.

Before we start, it's crucial to remember that simplifying an algebraic expression involves reducing it to its simplest form without changing its value. This often means factoring and canceling out common terms. So, let's get started with the first step: factoring the numerator.

Factoring the Numerator: 150c^2 - 500c

The first step to simplify this expression is to factor the numerator, which is 150c^2 - 500c. Factoring involves finding common factors in the terms of the expression and pulling them out. In this case, both terms have 'c' in them, and both are divisible by 50. So, we can factor out 50c from the numerator:

150c^2 - 500c = 50c(3c - 10)

Now, let's break down how we arrived at this factorization:

  • Identify common factors: Look at the coefficients (150 and 500) and the variable 'c'. The greatest common divisor (GCD) of 150 and 500 is 50. Also, both terms have 'c' as a common factor.
  • Factor out the common factors: Factor out 50c from each term.
    • 150c^2 divided by 50c is 3c.
    • -500c divided by 50c is -10.
  • Write the factored expression: Combine the common factor with the remaining terms in parentheses: 50c(3c - 10).

So, the factored form of the numerator 150c^2 - 500c is indeed 50c(3c - 10). Now that we have factored the numerator, we're ready to move on to the next step, which involves looking at the entire expression and seeing if we can cancel out any common factors between the numerator and the denominator.

Rewriting the Expression

Now that we have successfully factored the numerator, the next step is to rewrite the entire expression with the factored numerator. This will make it easier to see if we can simplify further. The original expression was:

(150c^2 - 500c) / (3c - 10)

We factored the numerator to get 50c(3c - 10). So, we can rewrite the expression as:

[50c(3c - 10)] / (3c - 10)

Rewriting the expression in this way highlights the common factor (3c - 10) in both the numerator and the denominator. This is a crucial step because it sets us up for the next action, which is to cancel out these common factors.

Think of it like simplifying a regular fraction. For instance, if you have 6/8, you can rewrite it as (2 * 3) / (2 * 4), and then you can cancel out the common factor of 2 to get 3/4. We’re doing the same thing here, but with algebraic expressions.

Canceling Common Factors

Now, let's cancel out the common factors. We have the expression:

[50c(3c - 10)] / (3c - 10)

Notice that the term (3c - 10) appears in both the numerator and the denominator. As long as (3c - 10) is not equal to zero, we can cancel it out. Canceling common factors is a fundamental step in simplifying rational expressions.

So, we can cancel (3c - 10) from the numerator and the denominator:

[50c * (3c - 10)] / (3c - 10) = 50c

By canceling the common factor, we have significantly simplified the expression. The result is simply 50c. It’s important to remember the condition that (3c - 10) should not be equal to zero, which means c should not be equal to 10/3. This condition ensures that we are not dividing by zero, which is undefined in mathematics.

The Simplified Expression

After factoring, rewriting, and canceling common factors, we've arrived at the simplified expression. The original expression was:

(150c^2 - 500c) / (3c - 10)

And after all the steps, it simplifies to:

50c

This means that (150c^2 - 500c) / (3c - 10) is equivalent to 50c, provided that c ≠ 10/3. This is a much simpler form, which is easier to work with in various mathematical contexts.

Verifying the Solution

To verify the solution, we can substitute a value for 'c' into both the original expression and the simplified expression to see if they yield the same result. Let's choose c = 2:

Original expression:

(150(2)^2 - 500(2)) / (3(2) - 10) = (150(4) - 1000) / (6 - 10) = (600 - 1000) / (-4) = -400 / -4 = 100

Simplified expression:

50(2) = 100

Both expressions give the same result, 100, which confirms that our simplification is correct for c = 2. It's always a good practice to verify your solution with a couple of different values to ensure accuracy.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes that you should avoid:

  • Dividing by zero: Always ensure that the denominator is not equal to zero. This is especially important when canceling out common factors.
  • Incorrect factoring: Make sure you factor correctly. Double-check your factored expressions by distributing back to see if you get the original expression.
  • Canceling terms instead of factors: You can only cancel common factors, not terms. For example, you cannot cancel '3c' in the expression (150c^2 - 500c) / (3c - 10) before factoring.
  • Forgetting to distribute: When factoring, remember to distribute the common factor to all terms inside the parentheses.

Avoiding these mistakes will help you simplify algebraic expressions accurately and efficiently.

Conclusion

Simplifying the expression (150c^2 - 500c) / (3c - 10) involves factoring, rewriting, and canceling common factors. By following these steps carefully, we found that the simplified expression is 50c, provided that c ≠ 10/3. Always remember to verify your solution and avoid common mistakes to ensure accuracy.

Understanding and mastering these techniques is crucial for success in algebra and beyond. Keep practicing, and you'll become more comfortable and confident in simplifying algebraic expressions.

For more information on simplifying algebraic expressions, you can check out resources like Khan Academy's Algebra Section.