Polynomials

Algebra

Polynomial Operations

Multiplication

Let then

Division

Let where is one of then

where and the degree of the degree of

Remainder Theorem

If polynomial is divide by the remainder will be . Using the Division Algorithm we get that plugging in we get .

Factor Theorem

If then is a factor of . This trivially follows from the remainder theorem as hence the factor theorem.

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Roots

Reality of Roots

Given a polynomial with all real coefficients, it can any combination of complex or real roots as long as the complex roots come in complex conjugate pairs

Linear Transformations of Roots

Given a polynomial in terms of with roots , you can find the polynomial in terms of with roots , , by using the equation , then rearranging for and plugging into the original polynomial and expanding and simplifying.

Vieta’s Formulae

Given a polynomial with roots then for any integer ,

This is more easily written as . This can be simply proved directly by considering and expanding out and comparing coefficients of identical powers of .

Rational Roots Theorem

Suppose has a rational zero , then . Proof Suppose and for some coprime .

2. Multiply yielding
4. ( trivially holds so assume ) Thus, . But is coprime to and therefore to , so must divide the remaining factor .
5. On the other hand, shifting the term to the right side and factoring out on the left side produces:
6. Reasoning as before, it follows that divides .

Irrationality Criterion

Any rational zero of a monic polynomial must be an integer, conversely any non-integer zero of a monic polynomial is irrational. Proof Let . If then by Rational Root Theorem, therefore