Partial-Fraction Decomposition
When adding two rational expressions, there has to be a common denominator. But what if you wanted to take apart a rational expression and rewrite it as a sum of ratios?
First, in order to break down the rational expression, the denominator must be a composite to factor. Then set the ratio equal to a sum of ratios, where the denominator is separated with unknown numerators. Next, multiply both sides by the common denominator and finally break down the constant and coefficiant parts to solve.
Example 1:
x2 - 2x = x(x - 2)Factor the denominator.
Set up the partial fraction.
Multiply by the common denominator.
11x - 14 = A(x - 2) + BxEliminate the denominators.
11x - 14 = Ax - 2A + BxUse the distributive property.
11x - 14 = x(A + B)- 2ARegroup terms with the variable x.
11 = A + B and -14 = -2ASet equal the coefficients and the constants seperately.
A = 7 and B = 4Solve for A then for B.
Rewrite as a sum of ratios.
x2 - 2x = x(x - 2)Factor the denominator.
Set up the partial fraction.
Multiply by the common denominator.
11x - 14 = A(x - 2) + BxEliminate the denominators.
11x - 14 = Ax - 2A + BxUse the distributive property.
11x - 14 = x(A + B)- 2ARegroup terms with the variable x.
11 = A + B and -14 = -2ASet equal the coefficients and the constants seperately.
A = 7 and B = 4Solve for A then for B.
Rewrite as a sum of ratios.
Example 2:
x2 - 1 = (x + 1)(x - 1)Factor the denominator.
Set up the partial fraction.
Multiply by the common denominator.
8x + 2 = A(x - 1) + B(x + 1)Eliminate the denominators.
8x + 2 = Ax - A + Bx + B Use the distributive property.
8x + 2 = x(A + B) + B - A Regroup terms with the variable x.
8 = A + B and 2 = B - ASet equal the coefficients and the constants seperately.
Solve, by adding the two equations together, and get 10 = 2B
Then, B = 5 and A = 3
Rewrite as a sum of ratios.
x2 - 1 = (x + 1)(x - 1)Factor the denominator.
Set up the partial fraction.
Multiply by the common denominator.
8x + 2 = A(x - 1) + B(x + 1)Eliminate the denominators.
8x + 2 = Ax - A + Bx + B Use the distributive property.
8x + 2 = x(A + B) + B - A Regroup terms with the variable x.
8 = A + B and 2 = B - ASet equal the coefficients and the constants seperately.
Solve, by adding the two equations together, and get 10 = 2B
Then, B = 5 and A = 3
Rewrite as a sum of ratios.
Example 3:
x2 - x - 12 = (x + 3)(x - 4)Factor the denominator.
Set up the partial fraction.
Multiply by the common denominator.
-2x - 41 = A(x - 4) + B(x + 3) Eliminate the denominators.
-2x - 41 = Ax - 4A + Bx + 3B Use the distributive property.
-2x - 41 = x(A + B) - 4A + 3BRegroup terms with the variable x.
-2 = A + B and -41 = -4A + 3B Set equal the coefficients and the constants seperately.
-41 = -4(-2 - B) + 3B Solve for A in the first equation and substitute into the second.
-41 = 8 + 7B Use distributive property and combine like terms.
-49 = 7BSubtract 8 on both sides.
-7 = B and A = 5Divide by 7, then solve for A
Rewrite as a sum of ratios.
x2 - x - 12 = (x + 3)(x - 4)Factor the denominator.
Set up the partial fraction.
Multiply by the common denominator.
-2x - 41 = A(x - 4) + B(x + 3) Eliminate the denominators.
-2x - 41 = Ax - 4A + Bx + 3B Use the distributive property.
-2x - 41 = x(A + B) - 4A + 3BRegroup terms with the variable x.
-2 = A + B and -41 = -4A + 3B Set equal the coefficients and the constants seperately.
-41 = -4(-2 - B) + 3B Solve for A in the first equation and substitute into the second.
-41 = 8 + 7B Use distributive property and combine like terms.
-49 = 7BSubtract 8 on both sides.
-7 = B and A = 5Divide by 7, then solve for A
Rewrite as a sum of ratios.
Example 4: When a factor repeats, each power must be represented.
Set up the partial fraction.
Multiply by the common denominator.
3x + 5 = A(x - 1) + B Eliminate the denominators.
3x + 5 = Ax - A + B Use the distributive property.
A = 3 and B - A = 5, therefore B = 8.
Rewrite as a sum of ratios.
Set up the partial fraction.
Multiply by the common denominator.
3x + 5 = A(x - 1) + B Eliminate the denominators.
3x + 5 = Ax - A + B Use the distributive property.
A = 3 and B - A = 5, therefore B = 8.
Rewrite as a sum of ratios.
Example 5: A solution with three terms.
Set up the partial fraction.
Multiply by the common denominator.
8x2 + 24x + 28 = A(x + 2)2 + Bx(x + 2) + Cx Eliminate the denominators.
8x2 + 24x + 28 = Ax2 + 4Ax + 4A + Bx2 + 2Bx + CxUse the distributive property.
8x2 + 24x + 28 = x2 (A + B) + x(4A + 2B + C) + 4ARegroup terms with x2 and x.
x2:8 = A + B; x:24 = 4A + 2B + C; and 28 = 4A Set equal the coefficients and the constants seperately.
Since 28 = 4A this means that A = 7.
And 8 = A + B means that B = 1.
Then 24 = 4A + 2B + C; substitute values for A and B and find that C = -6.
Rewrite as a sum of ratios.
Set up the partial fraction.
Multiply by the common denominator.
8x2 + 24x + 28 = A(x + 2)2 + Bx(x + 2) + Cx Eliminate the denominators.
8x2 + 24x + 28 = Ax2 + 4Ax + 4A + Bx2 + 2Bx + CxUse the distributive property.
8x2 + 24x + 28 = x2 (A + B) + x(4A + 2B + C) + 4ARegroup terms with x2 and x.
x2:8 = A + B; x:24 = 4A + 2B + C; and 28 = 4A Set equal the coefficients and the constants seperately.
Since 28 = 4A this means that A = 7.
And 8 = A + B means that B = 1.
Then 24 = 4A + 2B + C; substitute values for A and B and find that C = -6.
Rewrite as a sum of ratios.
|
Related Links: Math algebra Multiplying Rational Equations Simplifying Rational Expressions Solving Rational Equations Algebra Topics |
To link to this Partial-Fraction Decomposition page, copy the following code to your site: