Math Problem Statement
Solution
We are asked to perform the indicated operations for the given expression:
Step-by-Step Solution:
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Simplify the terms:
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The first term : Notice that is a difference of squares and can be factored: So the first term becomes:
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The second term remains unchanged.
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The third term simplifies to .
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Common Denominator:
To combine the first two fractions, find a common denominator. The first term has a denominator of , while the second term has a denominator of . The least common denominator (LCD) is .
Rewrite the second term to have this common denominator:
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Combine the first two terms:
Now, subtract the two fractions: Expand the numerator of the second term: So, the combined fraction is:
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Add the constant term:
Now add the constant to the fraction:
We can leave this as the final answer unless a further simplification is requested.
Final Answer:
Let me know if you'd like further details or clarifications!
Related Questions:
- How can we simplify rational expressions more effectively?
- What are common factoring techniques, like the difference of squares, and when do they apply?
- How do we find a common denominator for more complex fractions?
- How would this problem change if we were given numerical values for and ?
- How can we manipulate algebraic fractions involving polynomials?
Tip:
When dealing with expressions that include a difference of squares, remember that always factors into , which can simplify many algebraic operations.
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Math Problem Analysis
Mathematical Concepts
Algebra
Simplifying Rational Expressions
Factoring
Formulas
Difference of Squares: a^2 - b^2 = (a - b)(a + b)
Theorems
Factoring Theorem
Suitable Grade Level
Grades 10-12