GCSE · Algebra

Algebraic Fractions and Proof

Algebraic fractions follow exactly the same rules as number fractions: simplify by cancelling common factors, multiply straight across, and add over a common denominator. The second half of the page uses algebra to prove that something is always true, which no amount of checking with numbers can do.

GCSE subject content: know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent; use algebra to support arguments; disprove by counter-example; Higher: simplify and manipulate algebraic fractions; solve equations involving algebraic fractions; construct algebraic proofs

Before you start. You need number fractions (see Fractions, decimals and percentages): ⁶⁄₉ = ⅔ because 3 is a common factor; ⅓ + ¼ = ⁴⁄₁₂ + ³⁄₁₂ = ⁷⁄₁₂. You also need to factorise: common factors, x² + bx + c and the difference of two squares (see Double brackets and factorising quadratics).

1 · Simplifying algebraic fractions Higher

An algebraic fraction has letters in the numerator (top) or denominator (bottom), like (x² − 9)/(x + 3). To simplify, you can only cancel factors: things that multiply the whole top and the whole bottom. So factorise first, then cancel any bracket (or number, or letter) that appears as a factor of both.

Simplify

test it with x =

Cancel factors, never terms. In (x + 3)/3 the 3 on top is added, not multiplied, so it can't cancel: (x + 3)/3 ≠ x. Test with x = 6: (6 + 3)/3 = 3, not 6. Likewise (x² − 9)/(x + 3) is not x² − 3. If you can't see a bracket multiplying the whole top and the whole bottom, factorise until you can.

2 · Multiplying and dividing Higher

Just like numbers: to multiply, multiply the tops and multiply the bottoms; to divide, flip the second fraction and multiply ("keep, change, flip"). Factorise everything first, so you can cancel before multiplying out: it keeps the algebra small.

Work out

test it with x =

3 · Adding and subtracting Higher

To add or subtract fractions you need a common denominator. For number denominators use the lowest common multiple; for letter or bracket denominators, multiply them together. Multiply each numerator by whatever its denominator was multiplied by, then combine the numerators. Put brackets round each numerator: with subtraction, the minus applies to the whole second numerator.

Work out

test it with x =

4 · Solving equations with algebraic fractions Higher

In an equation you can get rid of the fractions altogether: multiply every term on both sides by the common denominator. What's left is a linear or quadratic equation you already know how to solve. Check your answers don't make any denominator zero.

Solve

x =

5 · Identity or equation?

An equation is true only for particular values: 2x + 1 = 7 only when x = 3. An identity is true for every value of x, because both sides are the same expression written differently: 2(x + 4) ≡ 2x + 8. The symbol ≡ means "is identically equal to". Graph both sides: for an identity the two graphs are the same graph; for an equation they only meet at the solutions.

x =

6 · Writing numbers with algebra

To prove things about all numbers of a type, write a general one with a letter. If n is any integer (whole number):

Type of number	Write it as	Why
any even number	2n	a multiple of 2
any odd number	2n + 1	one more than an even number
consecutive integers	n, n + 1, n + 2	each is one more than the last
consecutive even (or odd) numbers	2n, 2n + 2 (or 2n + 1, 2n + 3)	they go up in 2s
a multiple of 5	5n	5 × an integer
two different odd numbers	2m + 1 and 2n + 1	use two letters, or they'd be the same number

Even numbers are pairs; an odd number is pairs plus one left over. See why odd + odd is always even:

m = n =

7 · Proof and counter-examples

Checking examples is not a proof: a statement might work for the first million numbers and fail for the next. An algebraic proof Higher uses a general n, so it covers every case at once. Write the numbers with algebra, simplify, then explain in words why the result has the property you want (for example, "3(n + 1) is 3 × an integer, so it is a multiple of 3").

Prove that…

To disprove a statement you only need one example where it fails: a counter-example. Find one for each statement below.

Statement

n =

Worked examples

Simplify (x² − 9)/(x² + x − 6) Higher

Factorise the top (difference of two squares): (x + 3)(x − 3).

Factorise the bottom: product −6, sum 1: (x + 3)(x − 2).

Cancel the common factor (x + 3): (x − 3)/(x − 2).

Check with x = 4: (16 − 9)/(16 + 4 − 6) = 7/14 = ½, and (4 − 3)/(4 − 2) = ½ ✓

Write (x + 1)/3 + (x − 2)/4 as a single fraction Higher

Common denominator: 12.

Convert: 4(x + 1)/12 + 3(x − 2)/12.

Combine and expand: (4x + 4 + 3x − 6)/12 = (7x − 2)/12.

Write 5/(x − 2) − 3/(x + 1) as a single fraction Higher

Common denominator: (x − 2)(x + 1).

Numerators: 5(x + 1) − 3(x − 2) = 5x + 5 − 3x + 6. (The minus applies to both terms of 3(x − 2): −3 × −2 = +6.)

Answer: (2x + 11)/((x − 2)(x + 1)). Check x = 4: 5/2 − 3/5 = 1.9, and 19/(2 × 5) = 1.9 ✓

Solve 6/x − 6/(x + 1) = 1 Higher

Multiply every term by x(x + 1): 6(x + 1) − 6x = x(x + 1).

Simplify: 6 = x² + x, so x² + x − 6 = 0.

Factorise: (x + 3)(x − 2) = 0, so x = 2 or x = −3. Neither makes a denominator 0 ✓. Check x = 2: 3 − 2 = 1 ✓

Show that 3(x + 4) − 2(x − 1) ≡ x + 14

Expand: 3x + 12 − 2x + 2. (−2 × −1 = +2.)

Collect: x + 14, which is the right-hand side, so the identity holds for every x ✓

Prove that the sum of three consecutive integers is a multiple of 3 Higher

Let the integers be n, n + 1 and n + 2, where n is any integer.

Add: n + (n + 1) + (n + 2) = 3n + 3.

Factorise: 3n + 3 = 3(n + 1).

Conclude: n + 1 is an integer, so 3(n + 1) is a multiple of 3. ∎

Show that "n² + n + 41 is always prime" is false

Look for a counter-example. It works for n = 0 to 39, which is why it fools people.

Try n = 40: 40² + 40 + 41 = 1600 + 40 + 41 = 1681 = 41 × 41.

So 1681 is not prime, and one counter-example is enough to disprove the statement.

Quick check

What you've learned

Higher Simplify algebraic fractions by factorising, then cancelling common factors. Never cancel terms that are added.
Higher Multiply across; to divide, flip the second fraction and multiply. Factorise and cancel first.
Higher Add or subtract over a common denominator, with brackets round each numerator.
Higher In an equation, multiply every term by the common denominator to clear the fractions, then solve.
An identity (≡) is true for all values; an equation only for some. Show an identity by simplifying one side into the other.
Even: 2n; odd: 2n + 1; consecutive: n, n + 1, n + 2. Higher A proof uses these, simplifies, and explains why the result has the property.
One counter-example is enough to disprove a statement; examples alone never prove one.