(b+c-a)/(a-b)(a-c) + (c+a-b)/(b-c)(b-a) + (a-b-c)/(c-a)(c-b)

asked by guest
on Oct 24, 2024 at 12:18 am



You asked:

Evaluate the expression: \(\frac{b + c - a}{\left(a - b\right) \cdot \left(a - c\right)} + \frac{c + a - b}{\left(b - c\right) \cdot \left(b - a\right)} + \frac{a - b - c}{\left(c - a\right) \cdot \left(c - b\right)}\)

MathBot Answer:

Evaluated



\(\displaystyle \frac{b + c - a}{\left(a - b\right) \cdot \left(a - c\right)} + \frac{c + a - b}{\left(b - c\right) \cdot \left(b - a\right)} + \frac{a - b - c}{\left(c - a\right) \cdot \left(c - b\right)} = \frac{- a + b + c}{\left(a - b\right) \left(a - c\right)} + \frac{a - b - c}{\left(- a + c\right) \left(- b + c\right)} + \frac{a - b + c}{\left(- a + b\right) \left(b - c\right)} \)

Expanded

\[\frac{b + c - a}{\left(a - b\right) \cdot \left(a - c\right)} + \frac{c + a - b}{\left(b - c\right) \cdot \left(b - a\right)} + \frac{a - b - c}{\left(c - a\right) \cdot \left(c - b\right)} = \frac{a}{a b - a c - b c + c^{2}} + \frac{a}{- a b + a c + b^{2} - b c} - \frac{a}{a^{2} - a b - a c + b c} - \frac{b}{a b - a c - b c + c^{2}} - \frac{b}{- a b + a c + b^{2} - b c} + \frac{b}{a^{2} - a b - a c + b c} - \frac{c}{a b - a c - b c + c^{2}} + \frac{c}{- a b + a c + b^{2} - b c} + \frac{c}{a^{2} - a b - a c + b c}\]

Factored

\[\frac{b + c - a}{\left(a - b\right) \cdot \left(a - c\right)} + \frac{c + a - b}{\left(b - c\right) \cdot \left(b - a\right)} + \frac{a - b - c}{\left(c - a\right) \cdot \left(c - b\right)} = - \frac{2 b}{\left(a - c\right) \left(b - c\right)}\]

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