Question

Find the partial fraction decomposition for each rational expression. 5-2x / (x2 + 2)(x - 1)

Accepted Answer

Identify the form of the partial fraction decomposition. Since the denominator is $$ (x^2 + 2)(x - 1) $$, where $$ x^2 + 2  is an $$irreducible quadratic and $$ x - 1  is a $$linear factor, the decomposition will be of the form: $$ \frac{5 - 2x}{(x^2 + 2)(x - 1)} = \frac{Ax + B}{x^2 + 2} + \frac{C}{x - 1} $$ where $$ A $$, $$ B $$, and $$ C $$ are constants to be determined. Multiply both sides of the equation by the common denominator $$ (x^2 + 2)(x - 1)  to $$clear the fractions: $$ 5 - 2x = (Ax + B)(x - 1) + C(x^2 + 2) . $$This step eliminates the denominators and allows us to work with polynomials. Expand the right-hand side by distributing: $$ (Ax + B)(x - 1) = Ax^2 - Ax + Bx - B $$ and $$ C(x^2 + 2) = Cx^2 + 2C . $$Combine these to get: $$ 5 - 2x = (A + C)x^2 + (-A + B)x + (-B + 2C) . $$Equate the coefficients of corresponding powers of $$ x $$ from both sides. On the left, the polynomial is $$ 5 - 2x = 0x^2 - 2x + 5 . So$$, set up the system: $$ \begin{cases} A + C = 0 \ -A + B = -2 \ -B + 2C = 5 \end{cases} . $$Solve the system of equations for $$ A $$, $$ B $$, and $$ C . $$Once these constants are found, substitute them back into the partial fraction form $$ \frac{Ax + B}{x^2 + 2} + \frac{C}{x - 1}  to $$complete the decomposition.

Find the partial fraction decomposition for each rational expression. 5-2x / (x² + 2)(x - 1)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Partial Fraction Decomposition

Factoring the Denominator

Setting Up and Solving Equations for Coefficients