Partial Fraction Calculator

Table of Contents

• What Is a Partial Fraction Calculator?
• How to Use the Partial Fraction Calculator
• Why Split a Fraction into Partial Fractions?
• The Cover-Up Method in Four Steps
• Worked Example: (5x + 7) / ((x + 1)(x + 2))
• More Examples at a Glance
• When the Method Changes
• Common Mistakes to Avoid
• FAQs

Some algebraic fractions are too complicated to work with as they stand, yet they become easy once they are split into simpler pieces. That split is called partial fraction decomposition. A partial fraction calculator performs the split for you: enter the numbers from your expression, and it returns the values of A and B together with the steps that produced them, so you can check your own working or learn the method as you go.

What Is a Partial Fraction Calculator?

A partial fraction calculator takes a rational expression, meaning one polynomial divided by another, and rewrites it as a sum of smaller fractions. This page covers the most common textbook case: a numerator of the form px + q over a denominator that splits into two different linear factors, (x + a)(x + b). The answer has the form A/(x + a) + B/(x + b), and the whole job is to find A and B.

The tool uses the cover-up method, the fastest reliable way to find those constants by hand. Rather than only giving the final answer, the partial fraction calculator shows the substitution for each constant, so students can follow the logic and spot where their own attempt went wrong. If you first need to brush up on ordinary number fractions, our fraction calculator covers adding, subtracting, multiplying and dividing them.

How to Use the Partial Fraction Calculator

The partial fraction calculator asks for four numbers, taken straight from your expression. The first two come from the numerator: p is the coefficient of x, and q is the constant term. For the numerator 5x + 7, p is 5 and q is 7. If the numerator has no x term, p is 0. If it is just a constant, both the method and the tool still work.

The second pair comes from the denominator. Write each factor as (x + a) and (x + b), then enter a and b. Watch the signs carefully here. A factor written as (x − 3) means a = −3, because the tool expects the form (x + a). A factor of just x means a = 0.

Press the button and the tool returns the finished split, followed by the working. If the two factors are identical, it will tell you so, because repeated factors need an extra term and a slightly different setup, explained further down this page.

Why Split a Fraction into Partial Fractions?

The main reason is integration. An expression like (5x + 7) / ((x + 1)(x + 2)) has no simple integral in its combined form. Once split into 2/(x + 1) + 3/(x + 2), each piece integrates immediately into a logarithm. Exam questions in calculus courses rely on this constantly, which is why the technique is taught just before integration by parts and substitution are combined.

Partial fractions also appear in engineering. Laplace transforms, used to analyse electrical circuits and control systems, produce rational expressions that must be split before they can be converted back into time-based behaviour. The same algebra you practise for exams is doing real work there.

Finally, splitting can make evaluation and graphing easier. Two simple fractions reveal vertical asymptotes and behaviour near them more clearly than one combined expression. Reference works such as Wolfram MathWorld’s page on partial fraction decomposition set out the general theory for denominators of any degree.

The Cover-Up Method in Four Steps

Start by writing the split with unknown constants: (px + q) / ((x + a)(x + b)) = A/(x + a) + B/(x + b). Then multiply every term by the full denominator. The fractions clear, leaving px + q = A(x + b) + B(x + a).

To find A, choose the value of x that makes the first factor zero, namely x = −a. Substituting it removes the B term completely, because that term is multiplied by (x + a). This is the “cover up” the method is named after: you cover the factor, and what remains gives A directly after one division.

Repeat with x = −b to find B, this time covering the second factor. Two short substitutions give both constants, and no simultaneous equations are needed. The partial fraction calculator follows exactly these steps and prints each substitution, so its working matches what a teacher would write on a board.

Worked Example: (5x + 7) / ((x + 1)(x + 2))

Take the expression (5x + 7) / ((x + 1)(x + 2)). Here p = 5, q = 7, a = 1 and b = 2. The split must look like A/(x + 1) + B/(x + 2), and clearing the fractions gives 5x + 7 = A(x + 2) + B(x + 1).

For A, set x = −1 so the first factor vanishes: A = (5 × −1 + 7) / (−1 + 2) = 2 / 1 = 2. For B, set x = −2: B = (5 × −2 + 7) / (−2 + 1) = (−3) / (−1) = 3.

So the answer is 2/(x + 1) + 3/(x + 2). A quick check at x = 0 confirms it: the original expression gives 7/2 = 3.5, and the split gives 2/1 + 3/2 = 3.5 as well. The partial fraction calculator shows this same working when you enter p = 5, q = 7, a = 1 and b = 2, and checking an answer at a spare value of x is a habit worth building for your own work too.

More Examples at a Glance

The table below collects a few more splits produced by the same method. Trying each one in the tool and comparing the printed steps is a fast way to build confidence.

The last row shows that A and B do not have to be whole numbers. When they are fractions or decimals, the method is unchanged; only the arithmetic is slightly longer. The tool reports such values to four decimal places.

When the Method Changes

Three situations need extra care. First, repeated factors: if the denominator is (x + a)², the split needs two terms, A/(x + a) and B/(x + a)², and the cover-up method finds only one constant directly. The other comes from comparing coefficients or substituting a second value of x.

Second, improper expressions: if the numerator’s degree is not smaller than the denominator’s, divide first. Polynomial division reduces the expression to a polynomial plus a proper fraction, and only then does decomposition apply. Our long division calculator with steps demonstrates the division process with ordinary numbers, and the polynomial version follows the same rhythm.

Third, quadratic factors: a denominator factor like (x² + 1) that does not split into real linear factors takes a numerator of the form Ax + B instead of a single constant. The setup is a natural extension, but it is beyond the two-factor case this tool handles.

Common Mistakes to Avoid

The commonest error is a sign slip when reading the factors. Students see (x − 3) and enter a = 3, when the cover-up substitution needs x = 3, meaning a = −3 in the (x + a) form. Reading each factor aloud as “x plus something” before entering it prevents most of these mistakes.

Another frequent slip is substituting into the wrong line. The substitution x = −a must go into the cleared equation, px + q = A(x + b) + B(x + a), not into the original fraction, where it would divide by zero. Finally, always sanity-check the finished split at one value of x, as in the worked example. Ten seconds of checking catches nearly every arithmetic slip.

FAQs

What are partial fractions in simple words?

Partial fractions are the simpler fractions you get when one complicated algebraic fraction is split into a sum of smaller ones. Each smaller fraction has just one factor of the original denominator. The split form is easier to integrate, graph and evaluate than the combined form.

How does the cover-up method work?

You clear the fractions, then substitute the value of x that makes one factor zero. That wipes out one unknown term completely, leaving a one-step division for the other constant. Repeat for the second factor. The partial fraction calculator prints both substitutions so you can follow them.

Can the partial fraction calculator handle repeated factors?

Not directly. A repeated factor such as (x + 2)² needs an extra term, B/(x + 2)², alongside A/(x + 2), and only one of the constants comes from a cover-up substitution. The tool will tell you if you enter the same factor twice, so you know to switch to the extended method.

What if the numerator has no x term?

That is fine; it simply means p = 0. Enter 0 for p and the constant for q, and the method works as usual. A constant numerator over two linear factors is actually the simplest case of all, and it appears often in Laplace transform work.

Why do my A and B values come out as decimals?

Because the exact values are fractions, such as −2/3 and 5/3. The tool rounds to four decimal places for display. In exam work, keep the fractional form, since it is exact and usually simpler to integrate. Both forms represent the same split.

How can I check a partial fraction answer?

Pick a value of x that appears in neither factor’s root, such as x = 0, and evaluate both the original expression and your split. If the two numbers match, the split is almost certainly right. Testing a second value, such as x = 1, makes the check conclusive.

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