Math Problem Statement
Simplify the rational expression x 2 − 9 x 2 − x − 6
Solution
We are given the rational expression:
Step 1: Factor the numerator and denominator
Numerator:
This is a difference of squares, and it can be factored as:
Denominator:
We need to find two numbers that multiply to (the constant term) and add to (the coefficient of ). These numbers are and . So, the denominator factors as:
Step 2: Rewrite the rational expression
Now, substitute the factored forms of the numerator and denominator:
Step 3: Cancel out the common factor
Both the numerator and denominator have a common factor of . We can cancel this factor, assuming (since division by zero is undefined):
Final Answer:
The simplified expression is:
Note: This simplification is valid only for , since would make the original expression undefined.
Would you like any additional details or clarifications?
Here are some related questions you can explore:
- How do you factor quadratic expressions like ?
- What happens when you try to simplify expressions with different denominators?
- How does the concept of "difference of squares" help in factoring expressions?
- What is the meaning of canceling common factors in rational expressions?
- How do you deal with restrictions in rational expressions?
Tip: Always check for restrictions in rational expressions after simplifying. These restrictions are values that make the denominator zero, as division by zero is undefined.
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Math Problem Analysis
Mathematical Concepts
Algebra
Rational Expressions
Factoring
Formulas
Difference of squares: a^2 - b^2 = (a - b)(a + b)
Factoring quadratic trinomials: ax^2 + bx + c
Theorems
Factoring Theorem
Cancellation Law for Rational Expressions
Suitable Grade Level
Grades 9-10
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