Decompose (7x-x^2)^-1 in elementary fractions .

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll write the given expresison as a fraction, using the negative power property:

(7x-x^2)^-1 = 1/(7x-x^2)

We'll get 2 elementary fractions because we notice 2 factors at denominator.

1/(7x-x^2) = 1/x(7 - x)

The fraction 1/x(7 - x) is the result of algebraic addition of 2 elementary fractions, as it follows:

1/x(7 - x) = A/x + B/(7-x) (1)

We'll multiply by x(7 - x) both sides:

1 = A(7-x) + Bx

We'll remove the brackets:

1 = 7A - Ax + Bx

We'll factorize by x to the right side:

1 = x(B-A) + 7A

Comparing expressions of both sides, we'll get:

B-A = 0

A = B

7A = 1 => A = 1/7

B = 1/7

We'll replace A and B into the expression (1) and we'll get the result of decomposition into elementary fractions:

1/x(7 - x) = 1/7x + 1/(49-7x)

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