Find the value of a a-b a-c x-a - b b-c b-a x-b - c c-a c-b x-c -

a (a #8722;b)(a #8722;c)(x #8722;a) + b (b #8722;c)(b #8722;a)(x #8722;b) + c (c #8722;a)(c #8722;b)(x #8722;c) = a #10 ...

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Byju's Answer

Standard X

Mathematics

Addition of Rational Expressions

Find the valu...

Question

Find the value of $\frac{a}{(a - b) (a - c) (x - a)} + \frac{b}{(b - c) (b - a) (x - b)} + \frac{c}{(c - a) (c - b) (x - c)}$ .

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Solution

$\frac{a}{(a - b) (a - c) (x - a)} + \frac{b}{(b - c) (b - a) (x - b)} + \frac{c}{(c - a) (c - b) (x - c)}$

$=$ $\frac{a}{(a - b) (a - c) (x - a)} - \frac{b}{(b - c) (a - b) (x - b)} + \frac{c}{(c - a) (c - b) (x - c)}$

$=$ $\frac{a (b - c) (x - b) (x - c) - b (a - c) (x - a) (x - c) + c (a - b) (x - a) (x - b)}{(a - b) (a - c) (b - c) (x - a) (x - b) (x - c)}$
expanding numerator and denominator, we get

$=$ $\frac{(a b - a c) (x^{2} - x c - x b + b c) - (a b - b c) (x^{2} - x c - x a + a c) + (a c - b c) (x^{2} - x b - x a + a b)}{(a^{2} b - a^{2} c - a b c + a c^{2} - a b^{2} + a b c + b^{2} c - b c^{2}) (x - a) (x - b) (x - c)}$

$=$ $\frac{x (a^{2} b - a^{2} c + a c^{2} - a b^{2} + b^{2} c - b c^{2})}{(a^{2} b - a^{2} c + a c^{2} - a b^{2} + b^{2} c - b c^{2}) (x - a) (x - b) (x - c)}$

$=$ $\frac{x}{(x - a) (x - b) (x - c)}$

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Similar questions

Q. Prove that

\frac{(x - b) (x - c)}{(a - b) (a - c)} + \frac{(x - c) (x - a)}{(b - c) (b - a)} + \frac{(x - a) (x - b)}{(c - a) (c - b)} = 1

Q. Solve the equation

\frac{(x - b) (x - c)}{(a - b) (a - c)} + \frac{(x - c) (x - a)}{(b - c) (b - a)} + \frac{(x - a) (x - b)}{(c - a) (c - b)} = 1.

Q. The quadratic equation

\frac{(x + b) (x + c)}{(b - a) (c - a)} + \frac{(x + c) (x + a)}{(c - b) (a - b)} + \frac{(x + a) (x + b)}{(a - c) (b - c)} = 1

has

Q. Prove the identities:
(1)

\frac{a^{2} (x - b) (x - c)}{(a - b) (a - c)} + \frac{b^{2} (x - c) (x - a)}{(b - c) (b - a)} + \frac{c^{2} (x - a) (x - b)}{(c - a) (c - b)} = x^{2}

Let $ϕ (x) = \frac{(x - b) (x - c)}{(a - b) (a - c)} f (a) + \frac{(x - c) (x - a)}{(b - c) (b - a)} f (b) + \frac{(x - a) (x - b)}{(c - a) (c - b)} f (c) - f (x)$ Where a < c < b and $f^{11} (x)$ exists at all points in (a,b) . Then, there exists a real number $μ$ a < $μ$ < b such that $\frac{f (a)}{(a - b) (a - c)} + \frac{f (b)}{(b - c) (b - a)} + \frac{f (c)}{(c - a) (c - b)} =$

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Find the value of a(a−b)(a−c)(x−a)+b(b−c)(b−a)(x−b)+c(c−a)(c−b)(x−c).

Find the value of $\frac{a}{(a - b) (a - c) (x - a)} + \frac{b}{(b - c) (b - a) (x - b)} + \frac{c}{(c - a) (c - b) (x - c)}$ .