Question

In Exercises 16–24, write the partial fraction decomposition of each rational expression.3x/(x - $$2)(x^2 + 1)$$

Accepted Answer

Step 1: Recognize that the given rational expression is 3x / ((x - $$2)(x^2 + 1)). $$The denominator is already factored into two distinct terms: a linear factor (x - 2) and an irreducible quadratic factor $$(x^2 + 1). $$Step 2: Set up the partial fraction decomposition. For the linear factor (x - 2), assign a constant numerator A. For the irreducible quadratic factor $$(x^2 + 1$$), assign a linear numerator Bx + C. The decomposition will look like: 3x / ((x - $$2)(x^2 + 1)) = A / (x - 2) + (Bx + C$$) / $$(x^2 + 1). $$Step 3: Combine the right-hand side into a single fraction. To do this, find a common denominator, which is (x - $$2)(x^2 + 1). $$Rewrite the terms as: $$A(x^2 + 1) + (Bx + C)(x - 2) / ((x$$ - $$2)(x^2 + 1)). $$Step 4: Expand the numerator. Distribute A across $$(x^2 + 1) $$and distribute (Bx + C) across (x - 2). This gives: $$A(x^2) + A + Bx(x) - Bx(2) + C(x) - C(2). $$Combine like terms in the numerator. Step 5: Equate the numerator of the combined fraction to the numerator of the original fraction, 3x. This will give you a system of equations to solve for A, B, and C by matching coefficients of like terms $$(x^2$$, x, and the constant term). Solve this system to find the values of A, B, and C.

In Exercises 16–24, write the partial fraction decomposition of each rational expression.3x/(x - 2)(x^2 + 1)

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Concetti chiave

Rational Expressions

Partial Fraction Decomposition

Polynomial Factorization