Polynomial Operations and Theorems

Polynomial Basics

$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

Degree: Highest power of $x$ Leading coefficient: $a_n$

Long Division and Synthetic Division

Synthetic Division (for dividing by $x - c$)

Divide $2x^3 - 5x^2 + 3x - 1$ by $x - 2$:

Use $c = 2$: Bring down, multiply, add pattern.

Result: $2x^2 - x + 1$ remainder $1$

Key Theorems

Remainder Theorem

When $P(x)$ is divided by $(x - c)$, the remainder equals $P(c)$.

Factor Theorem

$(x - c)$ is a factor of $P(x)$ if and only if $P(c) = 0$.

Fundamental Theorem of Algebra

A polynomial of degree $n$ has exactly $n$ roots (counting multiplicity and complex roots).

Rational Root Theorem

Possible rational roots of $a_nx^n + \cdots + a_0$ are $\pm \frac{\text{factors of } a_0}{\text{factors of } a_n}$.

End Behavior

| Degree | Leading Coeff | Left End | Right End | |--------|--------------|----------|-----------| | Even | Positive | $\uparrow$ | $\uparrow$ | | Even | Negative | $\downarrow$ | $\downarrow$ | | Odd | Positive | $\downarrow$ | $\uparrow$ | | Odd | Negative | $\uparrow$ | $\downarrow$ |

Multiplicity of Zeros

• Odd multiplicity: Graph crosses the x-axis
• Even multiplicity: Graph touches and bounces off the x-axis

Example: $f(x) = (x-1)^2(x+3)$

• $x = 1$ (multiplicity 2): bounces
• $x = -3$ (multiplicity 1): crosses

Graphing strategy: Find zeros, determine end behavior, check multiplicity, plot a few extra points.