How do you express $\frac{6 x^{2} + 1}{x^{2} {(x - 1)}^{2}}$ $(6x^2+1)/(x^2(x-1)^2)$ in partial fractions?

Question

How do you express $\frac{6 x^{2} + 1}{x^{2} {(x - 1)}^{2}}$ $(6x^2+1)/(x^2(x-1)^2)$ in partial fractions?

1 Answer

Impact of this question

4239 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-25T20:33:39

$\frac{6 x^{2} + 1}{x^{2} {(x - 1)}^{2}} = \frac{2}{x} + \frac{1}{x^{2}} - \frac{2}{x - 1} + \frac{7}{{(x - 1)}^{2}}$ $(6x^2+1)/(x^2(x-1)^2) = 2/x+1/x^2-2/(x-1)+7/(x-1)^2$

Explanation:

$\frac{6 x^{2} + 1}{x^{2} {(x - 1)}^{2}} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x - 1} + \frac{D}{{(x - 1)}^{2}}$ $(6x^2+1)/(x^2(x-1)^2) = A/x+B/x^2+C/(x-1)+D/(x-1)^2$

So:

$6 x^{2} + 1 = A x {(x - 1)}^{2} + B {(x - 1)}^{2} + C x^{2} (x - 1) + D x^{2}$ $6x^2+1 = Ax(x-1)^2+B(x-1)^2+Cx^2(x-1)+Dx^2$

Putting $x = 1$ $x=1$ we get:

$7 = D$ $7 = D$

Putting $x = 0$ $x=0$ we get:

$1 = B$ $1 = B$

So:

$6 x^{2} + 1 = A x {(x - 1)}^{2} + {(x - 1)}^{2} + C x^{2} (x - 1) + 7 x^{2}$ $6x^2+1 = Ax(x-1)^2+(x-1)^2+Cx^2(x-1)+7x^2$

$6 x^{2} + 1 = A x (x^{2} - 2 x + 1) + (x^{2} - 2 x + 1) + C x^{2} (x - 1) + 7 x^{2}$ $color(white)(6x^2+1) = Ax(x^2-2x+1)+(x^2-2x+1)+Cx^2(x-1)+7x^2$

$6 x^{2} + 1 = A (x^{3} - 2 x^{2} + x) + (x^{2} - 2 x + 1) + C (x^{3} - x^{2}) + 7 x^{2}$ $color(white)(6x^2+1) = A(x^3-2x^2+x)+(x^2-2x+1)+C(x^3-x^2)+7x^2$

$6 x^{2} + 1 = (A + C) x^{3} + (8 - 2 A - C) x^{2} + (A - 2) x + 1$ $color(white)(6x^2+1) = (A+C)x^3+(8-2A-C)x^2+(A-2)x+1$

Hence:

$A = 2$ $A=2" "$ and $C = - 2$ $" "C = -2$

So:

$\frac{6 x^{2} + 1}{x^{2} {(x - 1)}^{2}} = \frac{2}{x} + \frac{1}{x^{2}} - \frac{2}{x - 1} + \frac{7}{{(x - 1)}^{2}}$ $(6x^2+1)/(x^2(x-1)^2) = 2/x+1/x^2-2/(x-1)+7/(x-1)^2$

Science

Math

Humanities

... and beyond

How do you express $\frac{6 x^{2} + 1}{x^{2} {(x - 1)}^{2}}$ $(6x^2+1)/(x^2(x-1)^2)$ in partial fractions?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you express 6x2+1x2(x−1)2(6x^2+1)/(x^2(x-1)^2) in partial fractions?

1 Answer

Explanation:

Related questions

Impact of this question

How do you express $\frac{6 x^{2} + 1}{x^{2} {(x - 1)}^{2}}$ $(6x^2+1)/(x^2(x-1)^2)$ in partial fractions?