U33 Polynomials: Essential Formulas

General Form of a Polynomial

A polynomial in $x$ of degree $n$ is an expression of the form $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$, where $a_n, a_{n-1}, \dots, a_0$ are real numbers and $a_n \neq 0$.

Here, $n$ is a non-negative integer. The coefficient $a_n$ is called the leading coefficient, and $a_0$ is the constant term.

Addition, Subtraction, and Multiplication

Polynomials are added, subtracted, and multiplied by combining like terms and applying the distributive law. For example, if $P(x)=2x^2+3x-1$ and $Q(x)=x^2-2x+4$, then $P(x)+Q(x)=3x^2+x+3$.

The Division Algorithm

For two polynomials $P(x)$ and $D(x)$ where $D(x) \neq 0$, there exist unique polynomials $Q(x)$ (quotient) and $R(x)$ (remainder) such that $P(x) = D(x) \cdot Q(x) + R(x)$, where the degree of $R(x)$ is less than the degree of $D(x)$.

Remainder Theorem

When a polynomial $P(x)$ is divided by a linear divisor $(x - a)$, the remainder is equal to $P(a)$.

Factor Theorem

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

Useful Factorizations

Common identities used in factoring polynomials:

Method of Comparing Coefficients

If two polynomials are equal for all values of $x$, then the coefficients of corresponding terms must be equal. This is a key technique for finding unknown constants.

Example: If $Ax^2 + Bx + C \equiv 3x^2 - 5$, then by comparing coefficients, we get $A=3$, $B=0$, and $C=-5$.