Algebraic fractions.
Use familiar fraction rules, while preserving factors and denominator restrictions.
Simplify rational expressions · add, subtract, multiply and divide · retain excluded values
Fractions with letters obey the same rules
An algebraic fraction is still a division. The denominator must not be zero, just as a numerical calculation cannot divide by zero. Before simplifying, identify values of the letter that would make the original denominator zero.
Cancellation removes common multiplicative factors. It cannot remove terms from a sum. Factorising numerator and denominator first exposes whole factors that can be divided out while preserving the original restrictions on the variable.
Keep denominators non-zero. Factor first, then cancel common factors, never separate terms within a sum.
Simplify x² − 2xx² − 5x + 6.
The cancelled factor still gives a restriction from the original denominator. The simplified expression must retain x ≠ 2 as well as x ≠ 3.
Pause and explainCan x be cancelled from (x + 3) divided by x?
No. The numerator is a sum, not a product with an overall factor x. Cancellation cannot remove just one part of that sum.
Add and subtract with a common denominator
Adding algebraic fractions follows the same idea as adding thirds and quarters: the denominators must name the same-sized parts. A common denominator gives both fractions a shared form.
Multiply each numerator by exactly the factor used to change its denominator. Keep the whole numerator in brackets when subtracting, because the minus sign applies to every term. Combine the numerators only after the denominators match, then simplify if possible.
Simplify 3x + 2x − 4.
Restrictions: x ≠ 0, 4.
Simplify 1x − 2 + x + 1x − 3.
For subtraction, put the entire second numerator in brackets before distributing the minus sign.
Pause and explainWhy does a minus sign before a bracket affect every term inside?
The whole numerator is being subtracted. For example, subtracting (x + 2) means subtracting x and subtracting 2.
Multiply and divide: factor and cancel
For multiplication, the two numerators form one product and the two denominators form another. This often exposes common factors that can be cancelled before expansion, keeping the work shorter.
Division asks for multiplication by the reciprocal of the divisor. Turn over the entire second fraction, not just a convenient part of it, then multiply. Include any restrictions needed to make the original fractions and the division itself defined.
Multiply 43a × 109a.
Divide 43a by 109a.
Invert the divisor, then multiply. The divisor must be defined and non-zero.
Simplify x² − 9x² − x − 6 × x − 3x + 3.
Original restrictions: x ≠ 3, −2, −3.
Pause and explainWhen dividing by a fraction, which fraction is inverted?
The divisor, which is the second fraction. The first fraction stays in its original orientation.
Your turn. Preserve restrictions.
Start by factorising and recording forbidden values. Then identify whether the operation needs a common denominator, direct multiplication, or multiplication by a reciprocal.
Leave expressions in factored form while cancelling. At the end, check one permitted numerical value in both the original and simplified expressions to catch sign or grouping errors.
Simplify x² − 16x² + 4x.
Reveal worked answer
Simplify 2x − 32x.
Reveal worked answer
Simplify xx + 1 ÷ x²x² − 1.