Refresher - Theorems

The following theorems form the basis of the theory of equations, and help answer some of these questions.

• Fundamental Theorem of Algebra
  If a polynomial ${\displaystyle P\left(x\right)}$ has positive degree and complex coefficients, then ${\displaystyle P\left(x\right)}$ has at least one complex zero.

• Factor Theorem
  A polynomial ${\displaystyle P\left(x\right)}$ has a factor ${\displaystyle (x-c)}$ if and only if ${\displaystyle f\left(c\right)=0}$.

• Complete Factorization Theorem for Polynomials
  If ${\displaystyle P\left(x\right)}$ is a polynomial of degree ${\displaystyle n>0}$, then there exist n complex numbers, ${\displaystyle c_1,c_c,\dots ,c_n}$
  such that
  ${\displaystyle P\left(x\right)=a(x-c_1)(x-c_2)\dots (x-c_n)}$.

• Conjugate Pair Zeros
  If a polynomial ${\displaystyle P\left(x\right)}$ of degree ${\displaystyle n>1}$ has real coefficients and if ${\displaystyle z=a+bi}$ with ${\displaystyle b\ne 0}$ is a complex zero of ${\displaystyle P\left(x\right)}$, then the conjugate ${\displaystyle \overline{z}=a-bi}$ is also a zero of ${\displaystyle P\left(x\right)}$.

• Rational Zeros of Polynomials
  If the polynomial ${\displaystyle P\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+a_1+a_0}$
  Has integer coefficients and if ${\displaystyle \frac{c}{d}}$ is a rational zero of ${\displaystyle P\left(x\right)}$ such that c and d have no common prime factor, then
  1. the numerator c of the zero is a factor of the constant term ${\displaystyle a_0}$
  2. the denominator d of the zero is a factor of the leading term ${\displaystyle a_n}$

• Division of Polynomials
  If ${\displaystyle f\left(x\right)}$ and ${\displaystyle P\left(x\right)}$ are polynomials and if ${\displaystyle P\left(x\right)\ne 0}$, then there is exists unique polynomials ${\displaystyle q\left(x\right)}$ and ${\displaystyle r\left(x\right)}$ such that ${\displaystyle f\left(x\right)=P\left(x\right)?q\left(x\right)+r\left(x\right)}$,
  Where either ${\displaystyle r\left(x\right)=0}$ or the degree of r(x)is less than the degree of ${\displaystyle P\left(x\right)}$. The polynomial ${\displaystyle q\left(x\right)}$is the quotient, and ${\displaystyle r\left(x\right)}$ is the reminder in the division of ${\displaystyle f\left(x\right)}$ by ${\displaystyle P\left(x\right)}$.

• A polynomial cannot have more real zeros than its degree.