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Factoring formulas

Even powers split as squares

x²ⁿ - a²ⁿ = (xⁿ - aⁿ)(xⁿ + aⁿ)
x²ⁿ - a²ⁿ=(xⁿ - aⁿ)(xⁿ + aⁿ)recognise the pattern, then replace it with the factor form

Because x²ⁿ and a²ⁿ are both square-like powers, they factor as a difference of squares.

Detailed notes

Formula Walkthrough

Power identities generalise the same factor patterns to larger exponents.

Step 1: Even powers are a difference of squares

If both powers are even, rewrite them as squares and then factor.

x2n - a2n = (xn)2 - (an)2
x2n - a2n = (xn - an)(xn + an)

Step 2: Odd power difference

For odd n, x - a is a factor. The remaining factor descends in powers of x and rises in powers of a.

xn - an = (x - a)(xn - 1 + axn - 2 + ... + an - 1)

Step 3: Odd power sum

For odd n, x + a is a factor. The remaining factor alternates signs.

xn + an = (x + a)(xn - 1 - axn - 2 + a2xn - 3 - ... + an - 1)

Step 4: Use the pattern carefully

First decide whether the expression is a sum or difference, then check whether the power is even or odd.

difference: x - a
sum with odd n: x + a