Proving a polynomial identity

Prove polynomial identities and use them to establish numerical relationships

Problem types

Adding, subtracting, and multiplying polynomials
Factoring over the complex numbers · Earlier idea
The difference of squares identity extends to a sum of squares over the complex numbers: x squared + 1 = (x + i)(x - i).
Factoring quadratic expressions · Earlier idea
Factoring a difference of squares or a perfect-square trinomial uses identities such as a² − b² = (a + b)(a − b). Read from either side, the same identity also computes products and squares quickly.
Why each equation step is valid · Useful idea
A polynomial identity is shown by a chain of steps that each keep the expression equal.
Structure of an expression · Useful idea
Seeing the structure of an expression, such as a difference of squares, reveals identities.
Equivalent expressions · Earlier idea
Equivalent expressions have equal values for every input, and a polynomial identity is such an equivalence.

Operations on rational expressions · See it applied
Identities such as the difference of squares factor a numerator or denominator so that a rational expression simplifies.
The binomial theorem · Extend the idea
The identities for (a + b)² and (a + b)³ are the cases n = 2 and n = 3 of the binomial expansion. The binomial theorem gives the coefficients for every n.
The conjugate of a complex number · See it applied
The difference of squares identity gives (a + bi)(a - bi) = a squared + b squared, the product of a number and its conjugate.

Pythagorean and angle-sum identities · Related idea
Algebraic identities such as the difference of squares are used to prove trigonometric identities.
Dividing polynomials · Shared foundations

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