How do you express $\frac{3}{(1 + x) (1 - 2 x)}$ $3/((1 + x)(1 - 2x))$ in partial fractions?

Question

What does partial-fraction decomposition mean?
What is the partial-fraction decomposition of $\frac{5 x + 7}{x^{2} + 4 x - 5}$ $(5x+7)/(x^2+4x-5)$ ?
What is the partial-fraction decomposition of $\frac{x + 11}{(x + 3) (x - 5)}$ $(x+11)/((x+3)(x-5))$ ?
What is the partial-fraction decomposition of $\frac{x^{2} + 2 x + 7}{x {(x - 1)}^{2}}$ $(x^2+2x+7)/(x(x-1)^2)$ ?
How do you write $\frac{2}{x^{3} - x^{2}}$ $2/(x^3-x^2)$ as a partial fraction decomposition?
How do you write $\frac{x^{4}}{{(x - 1)}^{3}}$ $x^4/(x-1)^3$ as a partial fraction decomposition?
How do you write $\frac{3 x}{(x + 2) (x - 1)}$ $(3x)/((x + 2)(x - 1))$ as a partial fraction decomposition?
How do you write the partial fraction decomposition of the rational expression $\frac{x^{2}}{x^{2} + x + 4}$ $x^2/ (x^2+x+4)$ ?
How do you write the partial fraction decomposition of the rational expression # (3x^2 + 12x -...
How do you write the partial fraction decomposition of the rational expression $\frac{1}{(x + 6) (x^{2} + 3)}$ $1/((x+6)(x^2+3))$ ?

1363 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-03-21T18:56:26

$\frac{3}{(1 + x) (1 - 2 x)} = \frac{1}{1 + x} + \frac{2}{1 - 2 x}$ $3/((1+x)(1-2x)) = 1/(1+x) +2/(1-2x)$

$\frac{3}{(1 + x) (1 - 2 x)} = \frac{A}{1 + x} + \frac{B}{1 - 2 x}$ $3/((1+x)(1-2x)) = A/(1+x) +B/(1-2x)$

$3 = A (1 - 2 x) + B ((1 + x)$ $3=A(1-2x)+B((1+x)$

$3 = A - 2 A x + B + B x$ $3=A-2Ax+B+Bx$

$A + B = 3, - 2 A + B = 0$ $A+B=3, -2A+B=0$

$B = 3 - A \to - 2 A + 3 - A = 0 \to - 3 A + 3 = 0$ $B=3-A->-2A+3-A=0->-3A+3=0$

$- 3 A = - 3, A = 1, B = 2$ $-3A=-3,A=1,B=2$

$\frac{3}{(1 + x) (1 - 2 x)} = \frac{1}{1 + x} + \frac{2}{1 - 2 x}$ $3/((1+x)(1-2x)) = 1/(1+x) +2/(1-2x)$

How do you express 3(1+x)(1−2x)3/((1 + x)(1 - 2x)) in partial fractions?