0580 · 2.3Algebraic fractions

Fraction principles and cancellation

0:00An algebraic fraction can look much more complicated than an ordinary number fraction. Yet both follow exactly the same principles. The challenge is recognising which parts are being added and which parts are being multiplied. By the end, you will understand how to combine algebraic fractions and simplify them without cancelling something you should keep. You already need three familiar skills: working with number fractions, expanding brackets, and factorising expressions.

0:24Factorising means rewriting an expression as a product of factors. For example, x² − 2x becomes x multiplied by the quantity x − 2. We will use those skills rather than reteach them in detail. An algebraic fraction is a fraction containing an algebraic expression in its numerator, its denominator, or both. The numerator is the expression above the fraction bar. The denominator is the expression below it. A rational expression can be written as a fraction of polynomials, including expressions involving powers such as x².

0:57The fraction bar means division, and it groups the entire expression above it. So x plus one over x minus two means the whole numerator divided by the whole denominator. A denominator must never equal zero. In this example, x cannot equal two, because division by zero is undefined. Changing the appearance of a fraction does not have to change its value. Multiplying its numerator and denominator by the same quantity produces an equivalent fraction, as long as that quantity is not zero.

1:23Dividing both by a common factor that is not zero also preserves its value. These two ideas explain common denominators and cancellation, so neither process needs to be an unexplained trick. Consider 6/9. We can write the numerator as 3 × 2 and the denominator as 3 × 3. Dividing both by their common factor of three gives 2/3. Cancellation is therefore a shorter way of showing division by a common factor.

1:51The important word is factor. A factor multiplies the rest of an expression, whereas a term is a part joined by addition or subtraction. In x + 2, x and two are terms. In x multiplied by the quantity x + 2, the first x is a factor of the whole expression. You cannot cancel the x in x plus two over x. That x is only one term in the numerator, not a factor of the whole numerator.

2:17At x equal to four, the fraction is 6/4, which simplifies to 3/2. Incorrect cancellation would give two, so it would change the value.

Adding and subtracting fractions

2:25When adding fractions, the denominator tells us the size of the parts being counted. Thirds and halves are different sizes, so their numerators cannot simply be added. We rewrite them using matching parts, such as sixths. Algebraic numerators do not change this reasoning; they simply describe how many of those parts we have. Our first worked example is x/3 plus the quantity x − 4/2. A convenient common denominator is six.

2:50Multiply the numerator and denominator of the first fraction by two, giving 2x/6. For the second fraction, multiply both parts by three, giving three times the quantity x − 4/6. With matching denominators, combine the numerators to give 2x + 3 times the quantity x − 4, all over six. Expanding the bracket gives 2x + 3x − 12, all over six. Collecting like terms produces 5x − 12, all over six.

3:18The denominator stays six because we are counting sixths throughout. Notice that we multiplied the entire numerator of the second fraction by three. Multiplying only its x term would change the fraction rather than create an equivalent one. Also, adding the denominators would not describe the same sized parts. A common denominator is the reason addition works, not an optional decoration in the working. For subtraction, consider two x over three minus three times the quantity x − 5/2.

3:46Again, use six as the common denominator. The first fraction becomes 4x/6. Multiplying both parts of the second fraction by three gives nine times the quantity x − 5/6. Combine these as 4x − 9 times the quantity x − 5, all over six. Expanding the subtraction gives 4x − 9x + 45, all over six. Collecting like terms gives 45 − 5x, all over six. The positive forty-five appears because we subtract the negative forty-five produced inside the bracket.

4:18This is a common place to lose a sign. The subtraction applies to the whole second numerator, not just its first term. Keeping that numerator grouped until you expand makes the operation clear. A reliable approach is to establish the common denominator first, then expand carefully, and finally collect like terms.

Multiplying and dividing fractions

4:35Multiplication works differently because we are not combining differently sized parts. Taking a fraction of another fraction multiplies the numerators and multiplies the denominators. For example, one half of one third is one sixth. Algebraic fractions follow that same rule, so multiplication does not require a common denominator. Consider 3a/4 × 9a/10. Write the product as 3a × 9 a, over 4 × 10. Multiplying the numerators gives 27a², because a × a is a².

5:07The denominator becomes forty, so the answer is 27a²/40. We could simplify common numerical factors before multiplying if any were available. Here, twenty-seven and forty have no common factor greater than one. Neither denominator contains a, so there is no algebraic factor to cancel. Seeing the same letter twice in the numerators means multiplication, not cancellation. Division means finding how many times one quantity fits into another. To divide by a fraction that is not zero, multiply by its reciprocal.

5:38The reciprocal reverses its numerator and denominator, and multiplying a fraction by its reciprocal gives one. This explains why reversing the divisor turns division into multiplication. Now divide 3a/4 by 9a/10. Replace division by multiplication by the reciprocal of the second fraction. This gives 3a/4 × 10/9 a. Multiplying across produces thirty a over thirty-six a, which simplifies to 5/6. For that cancellation, a must not equal zero. In the original question, a equal to zero would make the divisor zero, so the division would be undefined.

6:13The simplified answer does not display that restriction, but the original calculation still has it. Also, only the divisor is reversed; the first fraction remains unchanged.

Factorising to simplify fractions

6:23When a fraction contains sums, factorising often reveals what can genuinely cancel. Expanding does the opposite: it turns a product into separate terms. Before simplifying a rational expression, look for common factors and factorisable quadratics. You are trying to expose multiplication, because cancellation works on factors of entire numerators and denominators. Simplify the fraction whose numerator is x² − 2x, and whose denominator is x² − 5x + 6. Factorising the numerator gives x multiplied by the quantity x − 2.

6:57For the denominator, we need two numbers whose product is six and whose sum is negative five. Negative two and negative three give the factors x − 2 and x − 3. The fraction is now x multiplied by x minus two, over the product of x − 2 and x − 3. Divide the numerator and denominator by their common factor, x − 2. The simplified expression is x over x minus three.

7:21The original denominator excludes both two and three as values of x. The restriction at two remains even though the factor x − 2 has disappeared. Simplification preserves values wherever the original expression was defined; it does not repair division by zero. At three, the simplified denominator still visibly becomes zero. Keeping track of original denominators prevents us from giving the simplified expression a wider range of allowed values.

Variable denominators and exam methods

7:47Variable denominators use the same common denominator principle as numerical ones. We choose an expression containing the factors needed by both denominators. Then we multiply each numerator by whatever factor its denominator was missing. This creates equivalent fractions whose numerators can be combined, while their shared denominator stays unchanged. Consider one over the quantity x − 2, plus a second fraction: x + 1, over x − 3. Use the product of x − 2 and x − 3 as the common denominator.

8:17Multiplying both parts of the first fraction by x − 3 gives that expression as its numerator. The second numerator becomes x plus one multiplied by x minus two. Combine the numerators as x − 3 plus the product of x + 1 and x − 2. Expanding gives x − 3 + x² − 2x + x − 2. Collecting like terms leaves x² − 5. The answer is that numerator over the product of x − 2 and x − 3.

8:45The x terms cancel through addition here, which is different from cancelling a common factor across a fraction bar. Neither denominator factor is a factor of the complete numerator, so the fraction cannot simplify further. The original denominators exclude two and three. Leaving the denominator factorised makes its structure easier to see. These methods must work without a calculator on Paper two, and they also apply on calculator Paper four.

9:09Keep algebraic answers exact unless the question asks for a numerical approximation. There is no rounding to do in these examples. Show equivalent fractions, factorisation and intermediate working so your method can be followed. An exam-style question might ask you to simplify our quadratic fraction plus one over x minus three. Factorising the first numerator and denominator gives x times x minus two over the product of x − 2 and x − 3.

9:35Cancelling the common factor leaves x over x minus three. Both fractions now have the same denominator. Add the numerators to obtain x plus one over x minus three. No further common factor cancels, so this is the simplified answer. The original expression still excludes two and three. This approach avoids unnecessary expansion because factorising reveals a simpler route before we combine the fractions. The main mistakes to avoid are adding denominators, losing subtraction signs, and cancelling terms instead of factors.

10:04In division, reversing the first fraction or both fractions changes the calculation. For rational expressions, cancelling before factorising can hide an invalid step. Write the structure clearly, and check that each cancellation divides the whole numerator and denominator. Algebraic fractions are ordinary fractions with expressions in place of some numbers. Addition and subtraction need a common denominator because the parts must match. Multiplication combines numerators and denominators, while division uses the reciprocal of the divisor.

10:34Factorising reveals common factors, and cancellation preserves the value only where the original expression is defined.