Overview

Many olympiad algebra problems involve expressions of degree three, four, or higher. These expressions initially can seem impossible to simplify.

However, many higher degree expressions possess hidden structure. By recognising standard factorisations and important identities, we can often transform a complicated expression into something much more manageable.

These ideas appear a lot in AMC, AIME and olympiad algebra.

Motivation

Suppose we wish to factor

Expanding is clearly not a sensible approach.

Instead notice that

This immediately becomes a difference of squares:

Each factor can then be factored again using the cubic identities:

Rather than performing tedious computations, we can use structure to split the expression apart.

This is the main idea of higher power factorisation.

Difference of Powers

One of the most important factorisation formulas is

For every positive integer $n$,

The factor $(a-b)$ always appears.

Example

Factor

Applying the formula gives

This factorisation appears frequently in number theory and polynomial problems.

Sum of Powers (Odd Exponent)

When $n$ is odd, we also have a factorisation for $a^n + b^n$:

The signs in the second factor alternate.

The sum of cubes ($n=3$) is the most common special case, but $n=5$ and $n=7$ also appear in olympiad problems.

Note that when $n$ is even, $a^n + b^n$ does not factor over the integers in general.

Difference of Squares

The most fundamental factorisation is

Many higher power factorisations begin by repeatedly applying this identity.

Example

Factor

Apply difference of squares repeatedly:

and

and

As such

Difference and Sum of Cubes

Two of the most useful factorisations are

Difference of Cubes

Sum of Cubes

These are special cases of the difference and sum of powers formulas above, and they appear throughout olympiad algebra.

Example

Factor

Since

we get

Example

Factor

Using

we get

Fourth Powers

Expressions involving fourth powers often hide simpler structure.

For example,

Applying difference of squares again gives

This factorisation appears frequently in divisibility and polynomial problems.

The Symmetric Cubic Identity

One of the most important olympiad identities is

This is known as the symmetric cubic identity.

The second factor can also be written as

which shows that for real numbers, the second factor is always non negative. This means the sign of $a^3+b^3+c^3-3abc$ is determined entirely by the sign of $a+b+c$ (when $a,b,c$ are real and not all equal).

Proof

Consider

Expanding and collecting like terms leads to

Although the expansion is somewhat lengthy, it is completely straightforward.

The important fact to remember is the resulting identity rather than the expansion itself.

The Special Case $a+b+c=0$

A particularly useful consequence occurs when

Substituting into the symmetric cubic identity gives

Therefore

This is one of the most frequently used identities in olympiad algebra. Whenever a problem gives or implies that three quantities sum to zero, this identity should come to mind.

Worked Examples

Example 1

Given that $a+b+c=0$, evaluate

$a^3+b^3+c^3$.

Using the special case above,

No expansion is required.

Example 2

Factor $x^3+y^3+z^3-3xyz$.

Applying the symmetric cubic identity gives

Example 3

Given $x+y+z=0$, simplify

$x^3+y^3+z^3+5xyz$.

Since

the expression becomes

Example 4

Given $a+b+c=0$, simplify

$\frac{a^3+b^3+c^3}{abc}.$

Since $a+b+c=0$, we have $a^3+b^3+c^3 = 3abc$, so

Example 5

Show that $n^5 - n$ is divisible by $30$ for all integers $n$.

Factor using difference of powers:

Continuing:

Among any three consecutive integers $n-1, n, n+1$, one is divisible by $2$ and one by $3$, so $n^5 - n$ is divisible by $6$.

For divisibility by $5$, check all residues modulo $5$. If $n \equiv 0$ then $n \equiv 0$. If $n \equiv 1$ then $n-1 \equiv 0$. If $n \equiv 4$ then $n+1 \equiv 0$. If $n \equiv 2$ then $n^2+1 \equiv 0$. If $n \equiv 3$ then $n^2+1 \equiv 0$.

In every case, one of the factors is divisible by $5$. Since $\gcd(5,6)=1$, we conclude $30 \mid n^5 - n$.

Key Contest Insight

When working with high degree expressions, avoid expanding immediately.

Instead look for:

• difference of squares,
• sum or difference of cubes (or higher odd powers),
• repeated factorisations,
• factors of the form $a+b+c$,
• and opportunities to apply $a^3+b^3+c^3-3abc$.

Many olympiad problems are designed so that recognising the correct identity is the hardest step.

Strategy Tips

• Always search for common factorisation patterns before expanding.
• If an expression contains $a+b+c$, check whether the symmetric cubic identity may apply.
• Repeatedly factor differences of squares whenever possible.
• Look for opportunities to rewrite exponents:

• If you are given $a+b+c=0$, consider

Common Pitfalls

• Expanding large expressions unnecessarily.
• Forgetting the sign in the sum of cubes formula.
• Applying the symmetric cubic identity incorrectly.
• Missing repeated applications of difference of squares.
• Forgetting that

requires the condition

Practice Problems