Partial Fraction Decomposition (Linear Denominators)

Key Questions

• What does partial-fraction decomposition mean?

  Answer:

  Transforming a rational polynomial into a sum of simpler rational polynomials due to the factorization of the denominator.

  Explanation:

  Let me try and explain this to you. :)

  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

  What is a partial-fraction decomposition?

  We are all familiar with the process of adding and subtracting fractions, e.g.

  `1/(x+1) + 2/(x+3) = (x+3)/((x+1)(x+3)) + (2(x+1))/((x+1)(x+3)) = (3x + 5)/(x^2 + 4x + 3)`

  Now, partial-fraction decomposition does exactly the reverse thing.

  It takes a rational polynomial, so a fraction like

  `(3x + 5)/(x^2 + 4x + 3)`

  from my example, and tries to decompose it into a sum of "simpler" fractions (to be more precise: into a sum of fractions which denominators are factors of the original fractions's denominator.)

  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  How to compute a partial-fraction decomposition?

  1) Linear and unique factors

  Let's stick with my example:

  `(3x + 5)/(x^2 + 4x + 3)`

  The first thing to do is a always to find a complete factorization of the denominator:

  `x^2 + 4x + 3 = (x + 3)(x + 1)`

  Here, all the factors are linear and unique, this is the simple case.

  The goal is to to find `A`, `B` so that the following equation holds:

  `(3x + 5)/((x+3)(x+1)) = A/(x+3) + B / (x+1)`

  To do so, first we should multiply both sides with the denominator:

  `<=> 3x+ 5 = A ( x+1) + B ( x+3)`

  Now, in order to solve this equation, we need to "group" the `color(red)(x)` terms and the terms `color(blue)("without "x)` (more precisely: `x^0` terms):

  `color(red)(3x)+ color(blue)(5) = color(red)(A * x) + color(blue)(A) + color(red)(B * x) + color(blue)(3 B)`

  Thus, we can form an equation based on the "red" terms and an equation based on the "blue" terms:

  `{ (3 = A + B), (5 = A + 3B):}`

  The solution of this linear equation system is inded `A = 2` and `B = 1` which leads us to the following partial-fraction decomposition:

  `(3x + 5)/((x+3)(x+1)) = A/(x+3) + B / (x+1) = 2/(x+3) + 1/(x+1)`

  ++++++++++++++++++++++

  2) Non-unique factors

  The computation is more complicated in case that the complete factorization of the original denominator doesn't consist just of linear and unique factors.

  Just a short example:

  Let's say we could factorize our rational polynomial like follows:

  `x/(x^2(x+1)^2(x-1)`

  As the factors `x` and `x+1` occur twice in the denominator, we need to account for that by building fractions with increasing powers of those factors.
  In this case, we would need to find `A`, `B`, `C`, `D` and `E` so that

  `x/(x^2(x+1)^2(x-1)) = A/x + B/x^2 + C/(x+1) + D/(x+1)^2 + E/(x-1)`

  holds.

  ++++++++++++++++++++++

  3) Non-linear factors

  Last but not least, there might be non-linear factors in your factorized denominator.

  Example:

  `1/(x^3+x) = 1/(x(x^2+1))`

  Unfortunately, you can't factorize `x^2+1` further (at least, for real numbers).

  In this case, your partial-fraction decomposition needs to look like follows:
  Find `A`, `B`, `C` so that the following equation holds:

  `1/(x(x^2+1)) = A/x + (Bx+C)/(x^2+1)`

  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  When can a partial-fraction decomposition be useful?

  Finally, I'd like to note that a very common application of this method is integration of rational polynomials:

  If you need to compute

  `int (3x + 5)/(x^2 + 4x + 3) "d"x`

  and know that `1/(x+1) + 2/(x+3) = (3x + 5)/(x^2 + 4x + 3)` holds,

  it's easy to integrate by "splitting" the original fraction into a sum of simpler fractions and integrating each one of them:

  `int (3x + 5)/(x^2 + 4x + 3) "d"x = int 1/(x+1) "d"x + int 2/(x+3) "d"x`

  Lotusbluete · 2 · Nov 11 2015

• What does partial-fraction decomposition mean?
• What is the partial-fraction decomposition of `(5x+7)/(x^2+4x-5)`?
• What is the partial-fraction decomposition of `(x+11)/((x+3)(x-5))`?
• What is the partial-fraction decomposition of `(x^2+2x+7)/(x(x-1)^2)`?
• How do you write `2/(x^3-x^2)` as a partial fraction decomposition?
• How do you write `x^4/(x-1)^3` as a partial fraction decomposition?
• How do you write `(3x)/((x + 2)(x - 1))` as a partial fraction decomposition?
• How do you write the partial fraction decomposition of the rational expression `x^2/ (x^2+x+4)`?
• How do you write the partial fraction decomposition of the rational expression `(3x^2 + 12x - 20)/(x^4 - 8x^2 + 16)`?
• How do you write the partial fraction decomposition of the rational expression `1/((x+6)(x^2+3))`?
• How do you write the partial fraction decomposition of the rational expression `(x^3-x^2+1) / (x^4-x^3)`?
• How do you write the partial fraction decomposition of the rational expression `(2x^2-18x-12)/(x^3-4x)`?
• How do you write the partial fraction decomposition of the rational expression `(x^4+1)/(x^5+6 x^3)`?
• How do you write the partial fraction decomposition of the rational expression `6/(x^2-25)^2`?
• How do you write the partial fraction decomposition of the rational expression `(x^2+9)/(x^4-2x^2-8)`?
• How do you write the partial fraction decomposition of the rational expression `(9x^2 + 1)/(x^2(x − 2)^2)`?
• How do you write the partial fraction decomposition of the rational expression `2 / (x(4x-1))`?
• How do you write the partial fraction decomposition of the rational expression `(5x^2 - 25x + 28) / (x^2(x-7))`?
• How do you write the partial fraction decomposition of the rational expression `(y^2 + 1) / (y^3 - 1)`?
• How do you write the partial fraction decomposition of the rational expression `(6x)/(x^3-8)`?
• How do you write the partial fraction decomposition of the rational expression `(x^4+6)/(x^5+7x^3)`?
• How do you write the partial fraction decomposition of the rational expression `5/(x^2-1)^2`?
• How do you write the partial fraction decomposition of the rational expression `( x-2 ) / (x^2 + 4x + 3)`?
• How do you write the partial fraction decomposition of the rational expression `(6x+5) / (x+2) ^4`?
• How do you write the partial fraction decomposition of the rational expression `(x-1)/(x^3 +x)`?
• How do you write the partial fraction decomposition of the rational expression `(2x-2)/((x-5)(x-3))`?
• How do you write the partial fraction decomposition of the rational expression `x/((x+7)(x+8)(x+9))`?
• How do you write the partial fraction decomposition of the rational expression `(x^2 -111)/ (x^4- x^2- 72)`?
• How do you write the partial fraction decomposition of the rational expression `1 / ((x^2 + 1) (x^2 +4))`?
• How do you write the partial fraction decomposition of the rational expression `(x-5)/(x-2)^2`?
• How do you write the partial fraction decomposition of the rational expression `(2x-3) / (x-1)^2`?
• How do you write the partial fraction decomposition of the rational expression `1/(x^3-5x^2)`?
• How do you write the partial fraction decomposition of the rational expression `(x^2 + 1)/(x(x + 1)(x^2 + 2))`?
• How do you write the partial fraction decomposition of the rational expression `x^3/[x(x^2+2x+1)]`?
• How do you write the partial fraction decomposition of the rational expression `(x² + 2x-7) /( (x+3)(x-1)²)`?
• How do you write the partial fraction decomposition of the rational expression `z/(z^4-1)`?
• How do you write the partial fraction decomposition of the rational expression `(7x-2)/((x-3)^2(x+1))`?
• How do you write the partial fraction decomposition of the rational expression `(-3x^2 +3x - 5) / (3x^3 +x^2 + 6x +2)`?
• How do you write the partial fraction decomposition of the rational expression `x^2/(x+1)^3`?
• How do you write the partial fraction decomposition of the rational expression `x / (x² - x - 2)`?
• How do you write the partial fraction decomposition of the rational expression `x^2 / (x² + x + 2)`?
• How do you write the partial fraction decomposition of the rational expression `(x+4)/((x+1)(x-2)^2 )`?
• How do you write the partial fraction decomposition of the rational expression `1/(x^2+x+1)`?
• How do you write the partial fraction decomposition of the rational expression `(x^3 + x) /(x^2 + x + 1)^2`?
• How do you write the partial fraction decomposition of the rational expression `[((3x^2)+3x+12) / ((x-5)(x^2+9))]`?
• How do you write the partial fraction decomposition of the rational expression `10/[(x-1)(x^2+9)]`?
• How do you write the partial fraction decomposition of the rational expression `(x^3 - 6x^2 + 11x - 6) / (4x^3 - 28x^2 + 56x - 32)`?
• How do you write the partial fraction decomposition of the rational expression `(4x+4)/(x^2(x+2))`?
• How do you write the partial fraction decomposition of the rational expression `( 3x) /( x^3 − 2x^2 − x + 2)`?
• How do you write the partial fraction decomposition of the rational expression `(x^2 + 2) /( x^3 + 3x^2 + 3x + 1)`?
• How do you write the partial fraction decomposition of the rational expression `(x^2 − x + 1 )/( x^3 − x^2 + x − 1)`?
• How do you write the partial fraction decomposition of the rational expression `(-13x+11) / (2x^3 - 2x^2 + x - 1)`?
• How do you write the partial fraction decomposition of the rational expression `8 / [ ( x + 2 ) ( x^2 + 4 ) ]`?
• How do you write the partial fraction decomposition of the rational expression `(x+6)/(x^3-3x^2-4x+12)`?
• How do you write the partial fraction decomposition of the rational expression `(4x^2+ 21x-22)/((2x-1)(x^2 +x-42))`?
• How do you write the partial fraction decomposition of the rational expression `(s^3+s-4)/( (s^2+1)(s^2+4))`?
• How do you write the partial fraction decomposition of the rational expression `(3x^2+2x)/(x^2-4)`?
• How do you write the partial fraction decomposition of the rational expression `(x - 16) / (x^2 + x - 2)`?
• How do you write the partial fraction decomposition of the rational expression `3/(x ^(2) + 16x +5)`?
• How do you write the partial fraction decomposition of the rational expression `(2x^3-x^2+x+5)/(x^2+3x+2)`?
• How do you write the partial fraction decomposition of the rational expression `(4x+23)/(x^2-x-6)`?
• How do you write the partial fraction decomposition of the rational expression `(x+4)/[(x+1)^2 + 4]`?
• How do you write the partial fraction decomposition of the rational expression `(x^3 - 2) / (x^4 - 1)`?
• How do you write the partial fraction decomposition of the rational expression `(x^2 - 3x) / ((x-1)(x+2))`?
• How do you write the partial fraction decomposition of the rational expression `(x^3+2)/(x^2-x)`?
• How do you write the partial fraction decomposition of the rational expression `(x+1)/(x^2(x-2))`?
• How do you write the partial fraction decomposition of the rational expression `1/(x^3-6x^2+9x)`?
• How do you write the partial fraction decomposition of the rational expression `(x+10)/(x^2+2x-8)`?
• How do you write the partial fraction decomposition of the rational expression `(5x+7)/(x^2+4x-5)`?
• How do you write the partial fraction decomposition of the rational expression `1/((x^4)-6(x^3))`?
• How do you write the partial fraction decomposition of the rational expression `(8x-1)/(x^3 -1)`?
• How do you write the partial fraction decomposition of the rational expression `x/ (x^3-2x+x)`?
• How do you write the partial fraction decomposition of the rational expression `(2x - 3) / (x - 1)^2`?
• How do you write the partial fraction decomposition of the rational expression `3 / (x^2 - 3x)`?
• How do you write the partial fraction decomposition of the rational expression `5 / (x^2 + x - 6)`?
• How do you write the partial fraction decomposition of the rational expression `(x + 2) /( x(x - 4))`?
• How do you write the partial fraction decomposition of the rational expression `1/ [(x-1) ( x+ 1) ^2]`?
• How do you write the partial fraction decomposition of the rational expression `(x^3-x^2+1) / (x^4-x^3)`?
• How do you write the partial fraction decomposition of the rational expression `(9x^2+9x+40)/(x(x^2+5))`?
• How do you write the partial fraction decomposition of the rational expression `(7 x^2 - 7 x+12)/(x^4+x^2-2)`?
• How do you write the partial fraction decomposition of the rational expression `(x^2 + 5x - 7 )/( x^2 (x+ 1)^2)`?
• How do you write the partial fraction decomposition of the rational expression `x/((x-1)(x^2+4)`?
• How do you write the partial fraction decomposition of the rational expression `[(3x)/(x^2-6x+9)]`?
• How do you write the partial fraction decomposition of the rational expression `(4x-1 )/( x^2+3x-4)`?
• How do you write the partial fraction decomposition of the rational expression `x^2 /(x^2 +x+2)`?
• How do you write the partial fraction decomposition of the rational expression `(2x+1)/(4x^2+12x-7)`?
• How do you write the partial fraction decomposition of the rational expression `2 / (x^3 + 1)`?
• How do you write the partial fraction decomposition of the rational expression `(5x - 1) / ((x - 2)(x + 1))`?
• How do you write the partial fraction decomposition of the rational expression `(x^2)/(x+1)^3`?
• How do you write the partial fraction decomposition of the rational expression `(3x^2 + 10x -5)/ ( (x+1)^2(x-2) )`?
• How do you write the partial fraction decomposition of the rational expression `(x-2) /( x^2+4x+3)`?
• How do you write the partial fraction decomposition of the rational expression `1/((x+7)(x^2+9))`?
• How do you write the partial fraction decomposition of the rational expression `(5x^3 -9x+6)/(x^3-x^2)`?
• How do you write the partial fraction decomposition of the rational expression `(10x + 2)/ (x^3 - 5x^2 + x - 5)`?
• How do you write the partial fraction decomposition of the rational expression `(3x^2 -7x+1) / (x-1)^3`?
• How do you write the partial fraction decomposition of the rational expression `x/ (8x^2−6x+1)`?
• How do you write the partial fraction decomposition of the rational expression `x^2/((x-1)(x+2))`?
• How do you write the partial fraction decomposition of the rational expression `(3x^2 + 4x) / (x^2 +1)^2`?
• How do you write the partial fraction decomposition of the rational expression `(-3x^2-5x+18)/((x+1)(x^2+9))`?
• How do you write the partial fraction decomposition of the rational expression `7/(x^2-14x)`?
• How do you write the partial fraction decomposition of the rational expression `(4x^2+3)/((x-5)^3)`?
• How do you write the partial fraction decomposition of the rational expression `(x+1)/( (x^2 )*(x-1) )`?
• How do you write the partial fraction decomposition of the rational expression `x/(x^3-x^2-6x)`?
• How do you write the partial fraction decomposition of the rational expression `(x^2+8)/(x^2-5x+6)`?
• How do you write the partial fraction decomposition of the rational expression `(x+13)/(x^3+2x^2-5x-6)`?
• How do you write the partial fraction decomposition of the rational expression `(-4x-2)/((x-4)(x-7))`?
• How do you write the partial fraction decomposition of the rational expression `(7x^2 - 12x + 11) / (2x^3 - 5x^2 + x +2)`?
• How do you write the partial fraction decomposition of the rational expression `(9x) / (x^2 + 2x - 8)`?
• How do you write the partial fraction decomposition of the rational expression `(x^2 + 4)/ ((x^2 - 4)(x^2 +2))`?
• How do you write the partial fraction decomposition of the rational expression `(x^2+x)/((x+2)(x-1)^2)`?
• How do you write the partial fraction decomposition of the rational expression `(8x^2 - 4x - 8)/(x^4 + 2x^3)`?
• How do you write the partial fraction decomposition of the rational expression `(x^2)/(x+1)^3`?
• How do you write the partial fraction decomposition of the rational expression `(2x-6)/((x-1)(x-2)^2)`?
• How do you write the partial fraction decomposition of the rational expression `x^2 / ((x-1)^2 (x+1))`?
• How do you express `f(x) = (3x + 2) / [(x - 1)(x + 4)]` in partial fractions?
• How do you express `(1) / (x * ( x^2 - 1 )^2)` in partial fractions?
• How do you express `(3x)/((x + 2)(x - 1))` in partial fractions?
• How do you express `(x^5 + 1)/(x^6 - x^4)` in partial fractions?
• How do you express `1 / ((x+5)^2 (x-1))` in partial fractions?
• How do you express `x^4/(x-1)^3` in partial fractions?
• How do you express `(5x-1)/(x^2-x-2)` in partial fractions?
• How do you express `(3x+18)/(x^2+5x+4)` in partial fractions?
• How do you express `(x^2-1)/((x)*(x^2+1) )` in partial fractions?
• How do you express `x^3/(x^2 + 4x + 3 )` in partial fractions?
• How do you express `1/ (x^4 +1)` in partial fractions?
• How do you express `3/ (x^2-6x+8)` in partial fractions?
• How do you express `2/(x^3-x^2)` in partial fractions?
• How do you express `x^3/[x(x^2+2x+1)]` in partial fractions?
• How do you express `(x^3+x^2+x+2)/(x^4+x^2)` in partial fractions?
• How do you express `1/((x+7)(x^2+9))` in partial fractions?
• How do you express `(x+1)/(x^2 + 6x)` in partial fractions?
• How do you express `(2x^3 -x^2)/((x^2 +1)^2)` in partial fractions?
• How do you express `(3x )/ (x^2 * (x^2+1) )` in partial fractions?
• How do you express `1/(1+e^x)` in partial fractions?
• How do you express `x/((x-1)(x^2+4)` in partial fractions?
• How do you express `(x^2-2x+1)/(x-2)^3` in partial fractions?
• How do you express `(2x - 1) / [(x - 1)^3 (x - 2)]` in partial fractions?
• How do you express `(4x^3) / (x^3 + 2x^2 - x - 2)` in partial fractions?
• How do you express `(x^2 - 3x) / ((x-1)(x+2))` in partial fractions?
• How do you express `(2x+ 3)/(x^4-9x^2)` in partial fractions?
• How do you express `(x^3)/(x^3+1)` in partial fractions?
• How do you express `(2x^2+5)/(x^2+1)^2` in partial fractions?
• How do you express `(x-1)/(x^3 +x)` in partial fractions?
• How do you express `x/(16x^4-1)` in partial fractions?
• How do you express `(5x^2+3x-2)/(x^3+2x^2)` in partial fractions?
• How do you express `(4x-1)/( x^2(x-4))` in partial fractions?
• How do you express `(x-1)/(1+x^2)` in partial fractions?
• How do you express `(x^2+x)/((x+2)(x-1)^2)` in partial fractions?
• How do you express `(6x^2+1)/(x^2(x-1)^2)` in partial fractions?
• How do you express `1/(x^3 +4x)` in partial fractions?
• How do you express `[(3x)/(x^2-6x+9)]` in partial fractions?
• How do you express `(4x-2) /( 3(x-1)^2)` in partial fractions?
• How do you express `1/(x^3-1)` in partial fractions?
• How do you express `(6x^2+8x+30)/(x^3-27)` in partial fractions?
• How do you express `(x-1)/(x^3 +x^2)` in partial fractions?
• How do you express `(x^3-x^2+1) / (x^4-x^3)` in partial fractions?
• How do you express `(2(1-x-x^2))/(1-x^2)` in partial fractions?
• How do you express `(x^2 + 33)/(x^3 + x^2)` in partial fractions?
• How do you express `(-2x-3)/(x^2-x)` in partial fractions?
• How do you express `(x^3+1) / (x^2+4)` in partial fractions?
• How do you express `1/(4x^2 - 9)` in partial fractions?
• How do you express `(x^2) / ( x^2 + 3x +2)` in partial fractions?
• How do you express `(3x-1)/(x^2-x)` in partial fractions?
• How do you express `(x+13)/(x^3+2x^2-5x-6)` in partial fractions?
• How do you express `1/(x^2+x+1)` in partial fractions?
• How do you express `(2x - 1) / [(x - 1)^3 (x - 2)]` in partial fractions?
• How do you express `2 / (x^3 + 1)` in partial fractions?
• How do you express `1/[(x^3)-1]` in partial fractions?
• How do you express `(x^2)/(x+1)^3` in partial fractions?
• How do you express `(2x) / (4x^2 + 12x + 9)` in partial fractions?
• How do you express `(4x^2+6x-2)/((x-1)(x+1)^2)` in partial fractions?
• How do you express `x^2/(x^2 + x +2)` in partial fractions?
• How do you express `(8x-1)/(x^3 -1)` in partial fractions?
• How do you express `x/ (x^3-2x^2+x)` in partial fractions?
• How do you express `(x^3 - 5x + 2) / (x^2 - 8x + 15)` in partial fractions?
• How do you express `1/(x^4 - 16)` in partial fractions?
• How do you express `1/((x+6)(x^2+3))` in partial fractions?
• How do you express `(x^2-2x-1) / ((x-1)^2 (x^2+1))` in partial fractions?
• How do you express `(3x) / (x-3)^2` in partial fractions?
• How do you express `(x+2) / (x(x-4))` in partial fractions?
• How do you express `(x^(2) - 1) / (x(x^(2)+1))` in partial fractions?
• How do you express `(x) / (x^(3)-x^(2)-2x +2)` in partial fractions?
• How do you express `(x+4)/(x^2 + 2x + 5)` in partial fractions?
• How do you express `(9x)/(9x^2+3x-2)` in partial fractions?
• How do you express `1/(e^x(e^x+1))` in partial fractions?
• How do you express `1/[(1+x)(1-2x)]` in partial fractions?
• How do you express `(x+1 )/[ x^2(x-2)]` in partial fractions?
• How do you express `(-7x^2-15x-8)/((x+3)(x^2+4))` in partial fractions?
• How do you express `(x^3-x^2-5x) /( x^2-3x+2)` in partial fractions?
• How do you express `(x^2)/((x^4) - (2x^2) - 8)` in partial fractions?
• How do you express `(17x-50)/(x^(2)-6x+8)` in partial fractions?
• How do you express `1/(x^3-6x^2+9x)` in partial fractions?
• How do you express `(3x+2) / (x^(2)+3x-4)` in partial fractions?
• How do you express `(x^4+1)/(x^5+6 x^3)` in partial fractions?
• How do you express `6/(x^2-25)^2` in partial fractions?
• How do you express `(x^3 + 2) / (x^4 -16)` in partial fractions?
• How do you express `(6x^2+7x-6)/((x^2-4)(x+2))` in partial fractions?
• How do you express `(5x^2+7x-4)/(x^3+4x^2)` in partial fractions?
• How do you express `(3x^2 - 4x - 2) / [(x-1)(x-2)]` in partial fractions?
• How do you express `(11x-2) /( x^2 + x-6)` in partial fractions?
• How do you express `(3x)/(x^2+x-2)` in partial fractions?
• How do you express `( x-2 ) / (x^2 + 4x + 3)` in partial fractions?
• How do you express `(6x+5) / (x+2) ^4` in partial fractions?
• How do you express `(x+4)/((x+1)(x-2)^2)` in partial fractions?
• How do you express `(x^2-16x+9)/(x^4+10x^2+9)` in partial fractions?
• How do you express `(x^3 +1)/ (x^2 +3)` in partial fractions?
• How do you express `1/ (x^2 + 3x -4)` in partial fractions?
• How do you express `(5x - 4 ) / (x^2 -4x)` in partial fractions?
• How do you express `1/ [ (x-1)(x+2)(x-3)]` in partial fractions?
• How do you express `(x^2 - 4) / (x -1)` in partial fractions?
• How do you express `(2x^5 -x^3 -1) / (x^3 -4x)` in partial fractions?
• How do you express `(x^3+4)/[x^2+4]` in partial fractions?
• How do you express `10 / [(x-1)(x^2 + 9)]` in partial fractions?
• How do you express `(4x^5 + 2x^4 + 3x^3 - x^2 - x + 1) /( x (x^2 + 1))` in partial fractions?
• How do you express `x^2 / (x-1)^3` in partial fractions?
• How do you express `2 / (x(4x-1))` in partial fractions?
• How do you express `(5x^2 - 25x + 28) / (x^2(x-7))` in partial fractions?
• How do you express `(y^2 + 1) / (y^3 - 1)` in partial fractions?
• How do you express `(2x^3-x^2+x+5)/(x^2+3x+2)` in partial fractions?
• How do you express `1/((x²)(x² + 3x + 2))` in partial fractions?
• How do you express `(x+1)/( (x^2 )*(x-1) )` in partial fractions?
• How do you express `x/(x^3-x^2-6x)` in partial fractions?
• How do you express `(x^2+8)/(x^2-5x+6)` in partial fractions?
• How do you express `(x^2) / (x^2+x+2)` in partial fractions?
• How do you express `(2x-2)/((x-5)(x-3))` in partial fractions?
• How do you express `(x^4)/((x²+1)(x²-1))` in partial fractions?
• How do you express `(2x^4 + 9x^2 + x - 4)/(x^3 + 4x)` in partial fractions?
• How do you express `(x³ - 7x - 3)/(x² - x -2)` in partial fractions?
• How do you express `(x²+2) / (x+3)` in partial fractions?
• How do you express `[((3x^2)+3x+12) / ((x-5)(x^2+9))]` in partial fractions?
• How do you express `(x^2 - 8x + 44) / ((x + 2) (x - 2)^2)` in partial fractions?
• How do you express `(4x^2 - 7x - 12) / ((x) (x+2) (x-3))` in partial fractions?
• How do you express `(-2x-3)/(x^2-x)` in partial fractions?
• How do you express `(x^2 - 3x + 2) / (4x^3 + 11x^2)` in partial fractions?
• How do you express `(x² + 2x-7 )/( (x+3)(x-1)²)` in partial fractions?
• How do you express `3/((x-2)(x+1))` in partial fractions?
• How do you express `( x^2 + 2 ) / (x+2)` in partial fractions?
• How do you express `x/((x-1)(x+1))` in partial fractions?
• How do you express `(13x + 2)/( x^3 -1)` in partial fractions?
• How do you express `x+22 div x^2+2x-8` in partial fractions?
• How do you express `3/((1 + x)(1 - 2x))` in partial fractions?
• How do you express `((x^3+1)/(x^2+3))` in partial fractions?
• How do you express `(x^3 +x^2+2x+1) / [(x^2+1) (x^2+2)]` in partial fractions?
• How do you express `(3x+5)/((3x+2)(2x+1) )` in partial fractions?
• How do you express `(x^4+6)/(x^5+7x^3)` in partial fractions?
• How do you express `(2x^2+4x+12)/(x^2+7x+10)` in partial fractions?
• How do you express `(2x^2-18x-12)/(x^3-4x)` in partial fractions?
• How do you express `1 / (x^6 - x^3)` in partial fractions?
• How do you express `(x+4)/[(x+1)^2 + 4]` in partial fractions?
• How do you express `1/(x^3-4x)` in partial fractions?
• How do you express `(x^3)/((x^2)-1)` in partial fractions?
• How do you express `(4x^3-22x^2+33x-6)/((x-3)^3(x+2))` in partial fractions?
• How do you express `x/((x^2-1)(x-1))` in partial fractions?
• How do you express `(x+1)/[(x^2+1)^2(x^2)]` in partial fractions?
• How do you express `(x^2 + 2x) / (x^3 - x^2 + x - 1)` in partial fractions?
• How do you express `(x^3+x^2+2x+1)/((x^2+1)(x^2+2))` in partial fractions?
• How do you express `( 1+2x )/( (6x^2+1)(1-3x))` in partial fractions?
• How do you express `(x + 2)/((2x+7)(x+1))` in partial fractions?
• How do you express `(x-9)/[(x+5)(x-2)]` in partial fractions?
• How do you express `(-5s-36) / [ (s+2) (s^2+9) ]` in partial fractions?
• How do you express `x^2/(x^2+9)^2` in partial fractions?
• How do you express `(7x-2)/((x-3)^2(x+1))` in partial fractions?
• How do you write the fractional decomposition of `(5x^2 + 7x + 3)/(x^3 + x)`?
• How do you decompose `(3p-1)/(p^2-1)` into partial fractions?
• How do you decompose `(x-6)/(x^2-2x)` into partial fractions?
• How do you decompose `(5m-4)/(m^2-4)` into partial fractions?
• How do you decompose `(-4y)/(3y^2-4y+1)` into partial fractions?
• How do you solve `2/(x^2-16x+64)-8/(x-7)=(Ax+B)/(x^2-16x+63)`?
• How do you write the partial fraction decomposition of `(2x^3-x^2+x+5)/(x(x+1)^2)`?
• Question #31f48
• Find the partial fraction decomposition of `(x^3+x^2+x+2)/(x^4+x^2)`?
• Question #7fec7
• How to solve `sum_(n=1)^oo 1/((3n-2)(3n+1))` using partial fractions?
• Can someone explain me this please?
• How do you evaluate `\frac { ( x + 1) ^ { 2} } { ( x - 2) ( x - 1) ^ { 2} } = \frac { A } { x - 2} + \frac {B}{x-1}+\frac { c } { ( x - 1) ^ { 2} }`?
• How do I split this into partial fractions?