Quick Math Partial Fractions
Enter expression in this form
Example: (7x+1)/((x-1)(x+2))
Partial Fractions in Algebra
Partial fractions is a technique used to decompose a complex rational expression into a sum of simpler fractions. This is especially useful in calculus for integrating rational functions. The basic idea is to express a rational function as a sum of fractions whose denominators are the factors of the original denominator.
Steps for Decomposing into Partial Fractions:
1. Factor the Denominator: Factor the denominator of the rational expression into irreducible factors.
2. Set Up the Partial Fractions: Write the expression as a sum of fractions with unknown coefficients. The form of these fractions depends on the factors of the denominator:
– For each linear factor \((ax + b)\), use \(\frac{A}{ax + b}\).
– For each repeated linear factor \((ax + b)^n\), use \(\frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + \cdots + \frac{A_n}{(ax + b)^n}\).
– For each irreducible quadratic factor \((ax^2 + bx + c)\), use \(\frac{Ax + B}{ax^2 + bx + c}\).
3. Combine and Solve: Combine the partial fractions over a common denominator and solve for the unknown coefficients by equating the numerator to the original numerator.
Detailed Example:
Decompose \(\frac{7x + 1}{(x – 1)(x + 2)}\) into partial fractions.
1. Factor the Denominator:
– The denominator is already factored as \((x – 1)(x + 2)\)
2. Set Up the Partial Fractions:
\[
\frac{7x + 1}{(x – 1)(x + 2)} = \frac{A}{x – 1} + \frac{B}{x + 2}
\]
3. Combine and Solve:
\[
7x + 1 = A(x + 2) + B(x – 1)
\]
Expand and combine like terms:
\[
7x + 1 = Ax + 2A + Bx – B
\]
\[
7x + 1 = (A + B)x + (2A – B)
\]
Equate coefficients:
– For \(x\): \(A + B = 7\)
– For the constant term: \(2A – B = 1\)
Solve the system of equations:
1. \(A + B = 7\)
2. \(2A – B = 1\)
Add the two equations:
\[
(A + B) + (2A – B) = 7 + 1
\]
\[
3A = 8 \implies A = \frac{8}{3}
\]
Substitute \(A = \frac{8}{3}\) into \(A + B = 7\):
\[
\frac{8}{3} + B = 7 \implies B = 7 – \frac{8}{3} = \frac{21}{3} – \frac{8}{3} = \frac{13}{3}
\]
4. Write the Partial Fractions:
\[
\frac{7x + 1}{(x – 1)(x + 2)} = \frac{\frac{8}{3}}{x – 1} + \frac{\frac{13}{3}}{x + 2}
\]
Simplify:
\[
\frac{7x + 1}{(x – 1)(x + 2)} = \frac{8}{3(x – 1)} + \frac{13}{3(x + 2)}
\]
So, the partial fraction decomposition of \(\frac{7x + 1}{(x – 1)(x + 2)}\) is:
\[
\frac{7x + 1}{(x – 1)(x + 2)} = \frac{8}{3(x – 1)} + \frac{13}{3(x + 2)}
\]
