Polynomial Theorems

Division

Division of polynomials

P (x) = D (x) \cdot Q (x) + R (x)

degree of R < degree of D

Remainder theorem

P (x) \div (x - c) leaves remainder P (c)

Factor theorem

(x - c) is a factor of P (x) \Leftrightarrow P (c) = 0

Rational root theorem

a rational root \frac{p}{q} has p ∣ a_{0} and q ∣ a_{n}

P(x) = aₙxⁿ + … + a₀

Fundamental theorem of algebra

degree n \geq 1 means exactly n complex roots

counting repeats

Sum of roots

r_{1} + r_{2} + r_{3} = - \frac{b}{a}

Sum of products in pairs

r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1} = \frac{c}{a}

Product of roots

r_{1} r_{2} r_{3} = - \frac{d}{a}

Sum of roots

- \frac{a_{n - 1}}{a_{n}}

Product of roots

(- 1)^{n} \frac{a_{0}}{a_{n}}

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