Functions - Polynomial Functions (Factor and Remainder Theorems)

Polynomial Definition: A polynomial $P(x)$ of degree $n$ is defined as $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $a_n \neq 0$. Visually, the graph of a polynomial is a continuous, smooth curve without breaks or sharp corners. The degree $n$ determines the maximum number of $x$-intercepts and the general 'end behavior' of the graph as $x$ approaches infinity.

The Remainder Theorem: This theorem states that when a polynomial $P(x)$ is divided by a linear factor $(x - c)$, the remainder is simply the value of the polynomial evaluated at $c$, or $R = P(c)$. This allows for quick calculation of remainders without performing full algebraic long division.

The Factor Theorem: A specific case of the Remainder Theorem, it states that $(x - c)$ is a factor of $P(x)$ if and only if $P(c) = 0$. Visually, this means that if $(x - c)$ is a factor, the graph of $y = P(x)$ will have an $x$-intercept at the point $(c, 0)$.

Division Algorithm: Any polynomial $P(x)$ can be expressed in terms of a divisor $D(x)$, a quotient $Q(x)$, and a remainder $R(x)$ such that $P(x) = D(x)Q(x) + R(x)$. In the context of the Factor Theorem, if the remainder is zero, the divisor and quotient are both factors of the original polynomial.

Linear Divisors of the form $(ax - b)$: When a polynomial is divided by $(ax - b)$, the Remainder Theorem can be extended to state that the remainder is $P(\frac{b}{a})$. If $P(\frac{b}{a}) = 0$, then $(ax - b)$ is a factor of the polynomial.

Roots and Multiplicity: If $(x - c)^k$ is a factor of $P(x)$, then $c$ is a root of multiplicity $k$. Visually, if $k=1$, the graph crosses the $x$-axis at $c$. If $k$ is even (e.g., $k=2$), the graph is tangent to the $x$-axis at $c$ (it touches and turns back, forming a 'U' or inverted 'U' shape). If $k$ is odd and greater than 1, the graph flattens as it crosses the axis.

Finding Unknown Coefficients: By using given information about factors or remainders, we can create a system of linear equations to solve for unknown coefficients (like $k$ or $m$) within a polynomial expression. For example, knowing that $(x-1)$ and $(x+2)$ are factors provides two equations: $P(1)=0$ and $P(-2)=0$.