Simplify (x^3-36x)/(x^2-3x-10): A Step-by-Step Guide
Algebraic expressions can sometimes look intimidating, but with the right approach, they can be simplified to a more manageable form. This guide will walk you through the process of simplifying the expression (x^3 - 36x) / (x^2 - 3x - 10) step by step. By the end of this article, you'll not only have the simplified expression but also a solid understanding of the techniques involved. So, let's dive in!
1. Factoring the Numerator (x^3 - 36x)
When it comes to simplifying algebraic expressions, factoring is often the first and most crucial step. In this case, we'll begin by factoring the numerator, which is x^3 - 36x. Factoring involves breaking down an expression into its constituent parts, making it easier to identify common factors that can be canceled out.
Identifying Common Factors
The first thing to look for is a common factor that can be factored out from all terms in the expression. In our numerator, both terms, x^3 and -36x, have 'x' as a common factor. Let's factor out 'x'.
x^3 - 36x = x(x^2 - 36)
Recognizing the Difference of Squares
After factoring out 'x', we are left with (x^2 - 36) inside the parentheses. This expression is in the form of a difference of squares, which is a^2 - b^2. Recognizing this pattern is key because the difference of squares can be easily factored into (a - b)(a + b).
In our case, x^2 - 36 can be seen as x^2 - 6^2. Thus, we can factor it as follows:
x^2 - 36 = (x - 6)(x + 6)
Complete Factorization of the Numerator
Now, combining the initial factoring of 'x' with the difference of squares factorization, we get the fully factored form of the numerator:
x^3 - 36x = x(x - 6)(x + 6)
This completes the factorization of the numerator. By breaking it down into these factors, we've made it much easier to see if there are any common factors with the denominator that we can cancel out. Factoring is like dissecting the expression to reveal its hidden structure, paving the way for simplification.
2. Factoring the Denominator (x^2 - 3x - 10)
After successfully factoring the numerator, our next focus is on the denominator of the expression, which is x^2 - 3x - 10. Factoring the denominator is just as crucial as factoring the numerator because it allows us to identify common factors between the two, which can then be simplified. Let's walk through the process step by step.
Identifying the Correct Factors
To factor the quadratic expression x^2 - 3x - 10, we need to find two numbers that multiply to -10 (the constant term) and add up to -3 (the coefficient of the 'x' term). Let's list some factor pairs of -10:
- -1 and 10
- 1 and -10
- -2 and 5
- 2 and -5
Among these pairs, 2 and -5 add up to -3. Therefore, these are the numbers we'll use to factor the quadratic expression.
Factoring the Quadratic Expression
Now that we have identified the correct factors, we can rewrite the quadratic expression in factored form:
x^2 - 3x - 10 = (x + 2)(x - 5)
Verification
To ensure that we have factored the expression correctly, we can expand the factored form and check if it matches the original expression:
(x + 2)(x - 5) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10
Since the expanded form matches the original expression, we can be confident that our factorization is correct. By factoring the denominator, we have now set the stage for simplifying the entire expression by identifying and canceling out common factors between the numerator and the denominator. This step is essential for reducing the expression to its lowest terms.
3. Simplifying the Expression
Now that we have successfully factored both the numerator and the denominator, we can proceed to simplify the entire expression. Simplification involves identifying common factors between the numerator and the denominator and then canceling them out. This process reduces the expression to its lowest terms, making it easier to understand and work with. Let's see how it's done.
Writing the Factored Expression
First, let's write down the original expression with both the numerator and the denominator in their factored forms:
(x^3 - 36x) / (x^2 - 3x - 10) = [x(x - 6)(x + 6)] / [(x + 2)(x - 5)]
Identifying Common Factors
Next, we need to identify any factors that appear in both the numerator and the denominator. In this case, there are no common factors that can be directly canceled out. The factors in the numerator are x, (x - 6), and (x + 6), while the factors in the denominator are (x + 2) and (x - 5). Since there are no matching factors, we cannot simplify the expression any further through direct cancellation.
Checking for Further Simplifications
Even though there are no direct common factors, it's always a good idea to double-check if there are any algebraic manipulations we can perform to reveal hidden simplifications. However, in this particular expression, no such manipulations are apparent. The expression is already in its simplest form.
The Simplified Expression
Since we cannot simplify the expression any further, the simplified form of the given expression is:
[x(x - 6)(x + 6)] / [(x + 2)(x - 5)]
This means that the expression is already in its lowest terms. While it might seem a bit anticlimactic, recognizing when an expression is already simplified is a crucial skill in algebra. It prevents us from wasting time trying to simplify something that is already in its simplest form. In this case, our initial factoring was essential to confirm that no further simplification was possible.
4. State Restrictions on the Variable
When simplifying rational expressions, it's essential to state any restrictions on the variable. Restrictions occur when certain values of the variable would make the denominator of the expression equal to zero, which is undefined in mathematics. Let's identify the restrictions on the variable for our simplified expression.
Identifying Values That Make the Denominator Zero
To find the restrictions, we need to determine the values of 'x' that would make the denominator (x + 2)(x - 5) equal to zero. This occurs when either (x + 2) = 0 or (x - 5) = 0.
Solving these equations gives us:
- x + 2 = 0 => x = -2
- x - 5 = 0 => x = 5
Stating the Restrictions
Therefore, the restrictions on the variable 'x' are that it cannot be equal to -2 or 5. We can express this as:
x ≠-2, x ≠5
These restrictions are important because if 'x' were to take on either of these values, the original expression would be undefined due to division by zero. By stating these restrictions, we ensure that our simplified expression is mathematically valid for all other values of 'x'. This step completes the simplification process, providing a fully accurate and mathematically sound result.
Conclusion
In summary, we have successfully simplified the expression (x^3 - 36x) / (x^2 - 3x - 10) by factoring both the numerator and the denominator, identifying that there were no common factors to cancel, and stating the restrictions on the variable. The simplified expression is [x(x - 6)(x + 6)] / [(x + 2)(x - 5)], with the restrictions x ≠-2 and x ≠5. Understanding these steps will help you tackle similar algebraic simplification problems with confidence. Keep practicing, and you'll become more adept at recognizing patterns and applying the appropriate techniques!
For further learning and to deepen your understanding of algebraic expressions, visit Khan Academy's Algebra Section.
