[tex]\frac{x^2-25}{x^2+8 x+15} \cdot \frac{x^2+3 x}{x-5}[/tex]
Answer (2)
The problem requires simplifying a given rational expression. We factor the polynomials in the numerator and denominator, cancel out the common factors, and obtain the simplified expression. The steps are as follows:
Factor the numerator and denominator of each fraction.
Multiply the fractions.
Cancel out the common factors.
The simplified expression is x .
Explanation
Problem Analysis We are asked to simplify the expression x 2 + 8 x + 15 x 2 − 25 ⋅ x − 5 x 2 + 3 x and choose the correct answer from the provided options.
Factoring First, we factor the numerator and denominator of each fraction.
Factoring expressions We have x 2 − 25 = ( x − 5 ) ( x + 5 ) and x 2 + 8 x + 15 = ( x + 3 ) ( x + 5 ) . Also, x 2 + 3 x = x ( x + 3 ) .
Rewriting the expression So, the expression becomes ( x + 3 ) ( x + 5 ) ( x − 5 ) ( x + 5 ) ⋅ x − 5 x ( x + 3 ) .
Multiplying fractions Now, we multiply the two fractions: ( x + 3 ) ( x + 5 ) ( x − 5 ) ( x + 5 ) ⋅ x − 5 x ( x + 3 ) = ( x + 3 ) ( x + 5 ) ( x − 5 ) ( x − 5 ) ( x + 5 ) x ( x + 3 ) .
Canceling common factors We cancel out common factors in the numerator and the denominator. We can cancel ( x − 5 ) , ( x + 5 ) , and ( x + 3 ) .
Simplifying After canceling, we are left with x .
Final Answer Therefore, the simplified expression is x .
Examples
Simplifying rational expressions is a fundamental skill in algebra and is used in various real-world applications. For instance, when designing structures, engineers often use rational functions to model stress and strain. Simplifying these expressions allows them to analyze the behavior of the structure more efficiently and ensure its stability. Similarly, in economics, simplifying rational expressions can help in analyzing cost and revenue functions, leading to better decision-making in business.
The expression x 2 + 8 x + 15 x 2 − 25 ⋅ x − 5 x 2 + 3 x simplifies to x after factoring, multiplying, and canceling common factors.
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