Factoring xⁿ ± yⁿ

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

The difference of two equal positive integer powers always contains the factor $x-y$ . A sum of powers contains $x+y$ when the exponent is odd. These identities generalize familiar square and cube rules.

Visual guide

VISUAL GUIDEEven and odd powers produce different real roots

x⁴ − 1
x³ + 1

With y = 1, x⁴ − 1 has roots at ±1. The odd-power sum x³ + 1 has a root at −1. Compare the symmetry before choosing the difference or odd-sum identity.

Method

For integer $n\ge1$ ,

x^n-y^n=(x-y)\sum_{k=0}^{n-1}x^{n-1-k}y^k.

For odd $n$ , the corresponding sum factors with alternating signs in the second factor.

Worked example

For $n=5$ ,

x^5+y^5=(x+y)(x^4-x^3y+x^2y^2-xy^3+y^4).

On multiplication, every mixed term cancels. For $x^6-y^6$ , first use the difference of squares $(x^3-y^3)(x^3+y^3)$ , then apply both cube identities.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Which expression has x+1 as a factor?

x²+1

x⁵+1

x⁴+1

Hint 1 · Find a starting point

A polynomial has factor x+1 when its value at −1 is zero.

Hint 2 · Take the next step

An odd power of −1 is −1; an even power is 1.

Show the reasoning

Answer: x⁵+1

(−1)⁵+1 = 0, whereas both even-power expressions give 2.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

The same sum formula does not work for even exponents: substituting $x=-y$ into $x^2+y^2$ usually does not give zero.

Check your understanding

Factor $x^4-y^4$ .

Show answer

$(x-y)(x+y)(x^2+y^2)$ .

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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