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Basics

Polynomials

Common polynomial identities and factorizations including squares, cubes, and difference of powers.

Polynomial identities are the standard expansions and factorisations of sums, differences, and powers of two terms. Recognising them on sight replaces a lot of multiplying out or trial factoring with template matching. The cube and higher-power identities all share one shape: a linear factor (x ± a) times a longer polynomial whose coefficients alternate or stay constant based on the sign.

Difference of squares factors into conjugate binomials. The cross terms cancel because the +y and -y contributions to the x coefficient annihilate, leaving only the difference of pure squares. This is the standard identity for rationalising denominators and spotting hidden factorisations.

x2−y2=(x+y)(x−y)x^{2}-y^{2}=\left(x+y\right)\left(x-y\right)

Product of two linear binomials sharing a leading variable (the FOIL pattern). The middle coefficient is the sum of the constants and the trailing coefficient is their product. This is how you factor quadratics by inspecting sum-product pairs of the constant term.

(x+a)(x+b)=x2+(a+b)x+ab\left(x+a\right)\left(x+b\right)=x^{2}+\left(a+b\right)x+ab

Square of a sum. The cross term 2xy appears because the product is counted twice (once as xy and once as yx). It is the most commonly dropped piece when squaring a binomial by hand.

(a+y)2=x2+2xy+y2\left(a+y\right)^{2}=x^{2}+2xy+y^{2}

Square of a difference. Identical to the sum case except the cross term picks up a negative sign from the negative second term. Squaring kills the sign on the y2 term but not on the cross term.

(a−y)2=x2−2xy+y2\left(a-y\right)^{2}=x^{2}-2xy+y^{2}

Cube of a sum. The coefficients 1, 3, 3, 1 are the fourth row of Pascal’s triangle, generalised by the binomial theorem to (x + y)n. Memorise the cube directly; re-deriving it each time is slower.

(a+y)3=x3+3xy2+3x2y+y3\left(a+y\right)^{3}=x^{3}+3xy^{2}+3x^{2}y+y{3}

Cube of a difference. Signs alternate because each occurrence of the negative term flips the sign. The x3 term stays positive because (-y)0 is positive; the y3 term is negative because (-y)3 is negative.

(a−y)3=x3−3xy2−3x2y−y3\left(a-y\right)^{3}=x^{3}-3xy^{2}-3x^{2}y-y{3}

Sum of cubes factors into a linear and a quadratic factor. The quadratic x2 - xy + y2 is irreducible over the reals (its discriminant is -3y2), so this is as far as the factorisation goes without complex numbers.

x3+y3=(x+y)(x2−xy+y2)x^{3}+y^{3}=\left(x+y\right)\left(x^{2}-xy+y^{2}\right)

Difference of cubes factors the same way, with the sign of the middle term of the quadratic reversed. Together with the sum-of-cubes formula, this is the n = 3 case of the general xn ± an identities below.

x3−y3=(x−y)(x2+xy+y2)x^{3}-y^{3}=\left(x-y\right)\left(x^{2}+xy+y^{2}\right)

Difference of even powers factors by treating xn and yn as a difference of squares. Iterate this halving until the exponent is odd or one to get a complete real factorisation, alternating sum and difference factors at each level.

x2x−y2x=(xn−yn)(xn+yn)x^{2x}-y^{2x}=\left(x^{n}-y^{n}\right)\left(x^{n}+y^{n}\right)

For odd n, both the difference and the sum of n-th powers factor cleanly. The difference identity holds for all n. The sum identity is restricted to odd n because xn + an has x = -a as a root only when raising -a to n recovers -an.

if n is odd thenif\text{ }n \text{ }is \text{ }odd \text{ }then

Difference of n-th powers (any positive integer n). The long factor is a geometric sum Σ xn-1-kak and reduces to (xn - an)/(x - a) when x ≠ a. The geometric series sum follows from this identity.

xn−an=(x−a)(xn−1+axn−2+...+an−1)x^{n}-a^{n}=\left(x-a\right)\left(x^{n-1}+ax^{n-2}+...+a^{n-1}\right)

Sum of n-th powers, for odd n, factors with alternating signs in the long factor. Substituting -a for a in the difference formula flips the sign on every other term, producing the alternation.

xn+an=(x+a)(xn−1−axn−2+a2xn−2+...+an−1)x^{n}+a^{n}=\left(x+a\right)\left(x^{n-1}-ax^{n-2}+a^{2}x^{n-2}+...+a^{n-1}\right)