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((a^(2))/(x-a)+(b^(2))/(x-b)+(c^(2))/(x-c)+a+b+c)/((a)/(x-a)+(b)/(x-b)+(c)/(x-c))

5 min read Jun 16, 2024
((a^(2))/(x-a)+(b^(2))/(x-b)+(c^(2))/(x-c)+a+b+c)/((a)/(x-a)+(b)/(x-b)+(c)/(x-c))

Simplifying a Complex Expression: ((a^(2))/(x-a)+(b^(2))/(x-b)+(c^(2))/(x-c)+a+b+c)/((a)/(x-a)+(b)/(x-b)+(c)/(x-c))

This article will guide you through the process of simplifying the complex algebraic expression:

((a^(2))/(x-a)+(b^(2))/(x-b)+(c^(2))/(x-c)+a+b+c)/((a)/(x-a)+(b)/(x-b)+(c)/(x-c))

Understanding the Expression

The expression involves several fractions with variables in the denominator, making it appear intimidating. However, the key to simplifying it lies in finding a common denominator and performing the necessary operations.

Step-by-Step Simplification

  1. Finding a Common Denominator:

    The least common denominator for all the fractions is (x-a)(x-b)(x-c). We multiply each term in the numerator and denominator by the missing factors to achieve this common denominator:

    Numerator:

    • (a^(2))/(x-a) * (x-b)(x-c)/(x-b)(x-c) = a^(2)(x-b)(x-c) / (x-a)(x-b)(x-c)
    • (b^(2))/(x-b) * (x-a)(x-c)/(x-a)(x-c) = b^(2)(x-a)(x-c) / (x-a)(x-b)(x-c)
    • (c^(2))/(x-c) * (x-a)(x-b)/(x-a)(x-b) = c^(2)(x-a)(x-b) / (x-a)(x-b)(x-c)
    • a * (x-a)(x-b)(x-c) / (x-a)(x-b)(x-c)
    • b * (x-a)(x-b)(x-c) / (x-a)(x-b)(x-c)
    • c * (x-a)(x-b)(x-c) / (x-a)(x-b)(x-c)

    Denominator:

    • (a)/(x-a) * (x-b)(x-c)/(x-b)(x-c) = a(x-b)(x-c) / (x-a)(x-b)(x-c)
    • (b)/(x-b) * (x-a)(x-c)/(x-a)(x-c) = b(x-a)(x-c) / (x-a)(x-b)(x-c)
    • (c)/(x-c) * (x-a)(x-b)/(x-a)(x-b) = c(x-a)(x-b) / (x-a)(x-b)(x-c)

  2. Combining Terms:

    Now that all terms have the same denominator, we can combine the numerators and denominators:

    Numerator: a^(2)(x-b)(x-c) + b^(2)(x-a)(x-c) + c^(2)(x-a)(x-b) + a(x-a)(x-b)(x-c) + b(x-a)(x-b)(x-c) + c(x-a)(x-b)(x-c)

    Denominator: a(x-b)(x-c) + b(x-a)(x-c) + c(x-a)(x-b)

  3. Factoring:

    We can factor out (x-a)(x-b)(x-c) from both the numerator and denominator:

    Numerator: (x-a)(x-b)(x-c) [a^(2) + b^(2) + c^(2) + a(x-b) + b(x-a) + c(x-c)]

    Denominator: (x-a)(x-b)(x-c) [a + b + c]

  4. Simplifying:

    The common factors (x-a)(x-b)(x-c) cancel out, leaving us with:

    (a^(2) + b^(2) + c^(2) + a(x-b) + b(x-a) + c(x-c)) / (a + b + c)

Final Result

Therefore, the simplified form of the given expression is: (a^(2) + b^(2) + c^(2) + a(x-b) + b(x-a) + c(x-c)) / (a + b + c)

This simplified expression is easier to understand and work with, highlighting the importance of algebraic manipulation techniques to make complex expressions more manageable.

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