Polynomials and Polynomial Functions

The zeros (roots) of a polynomial function are found by setting each factor equal to $0$ .
Multiplicity is determined by the exponent of each factor:
- An odd multiplicity indicates the graph crosses the $x$ -axis at that intercept.
- An even multiplicity indicates the graph touches and turns around (bounces) at that intercept.
Factored form functions can be written directly from identified roots and graph behaviors, for instance: $f(x) = x(x+4)(x-7)$ .

Complex roots of polynomials with real coefficients occur in conjugate pairs (e.g., if $x = 7i$ is a root, $x = -7i$ must also be a root, resulting in a quadratic factor of $(x - 7i)(x + 7i) = x^2 + 49$ ).
Polynomials of degree $3$ or $4$ can be solved by factoring out known linear or quadratic factors via polynomial division, followed by factoring the quotient using grouping, square roots, or completing the square:
- Completing the square is applied when quadratic factors are non-factorable over integers (e.g., $x^2 - 2x - 5 = 0 \implies x = 1 \pm \sqrt{6}$ ).
- Factoring by grouping is applied to four-term polynomials (e.g., $3x^3 - 5x^2 + 36x - 60 = (3x - 5)(x^2 + 12)$ , yielding zeros $x = \frac{5}{3}$ and $x = \pm 2i\sqrt{3}$ ).

Polynomial Long Division and Synthetic Division are methods used to simplify rational expressions and determine factors.
Factor Theorem: A polynomial divisor $d(x)$ is a factor of $f(x)$ if and only if the remainder of the division $\frac{f(x)}{d(x)}$ is equal to $0$ .
Remainder Theorem: Dividing a polynomial $f(x)$ by a linear binomial $(x - c)$ results in a remainder equal to $f(c)$ .
If evaluating $f(c)$ yields $0$ , $(x - c)$ is guaranteed to be a factor without requiring full division.

Polynomial operations involve combining like terms for addition and subtraction, or distributing terms across polynomials for multiplication.
Binomial expansions $(x + a)^n$ follow coefficients derived from Pascal's Triangle:
- Degree $0$ : $1$
- Degree $1$ : $1, 1$
- Degree $2$ : $1, 2, 1$
- Degree $3$ : $1, 3, 3, 1$