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Polynomial Summary

Polynomial

A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, $3 x^{2} + 2 x - 5$ is a polynomial.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $4 x^{3} + 3 x^{2} + 2 x + 1$ , the degree is 3.

Multiplication of Polynomials

To multiply two polynomials, each term in the first polynomial is multiplied by each term in the second polynomial. For example:

(2x + 3)(x - 4)	=	(2x * x) + (2x * (-4)) + (3 * x) + (3 * (-4))
	=	2x² - 8x + 3x - 12
	=	2x² - 5x - 12

(2x + 3)(x - 4) =

Multiply the term 2x with x.

Division of Polynomials

To divide polynomials, we use polynomial long division or synthetic division. For example, dividing $2 x^{3} + 3 x^{2} - x - 5$ by $x - 1$ :

$\begin{array}{rrrrr} x - 1 & 2 x^{3} & + 3 x^{2} & - x & - 5 \\ 2 x^{2} & + 5 x & + 4 \end{array}$

x-intercept

The x-intercepts of a polynomial are the points where the graph of the polynomial crosses the x-axis. These are the solutions to the equation $P (x) = 0$ .

y-intercept

The y-intercept of a polynomial is the point where the graph of the polynomial crosses the y-axis. This is the value of the polynomial when $x = 0$ .

Zero of Polynomial

The zeros of a polynomial are the values of $x$ for which the polynomial equals zero. These are also the x-intercepts of the polynomial.

Factor of Polynomial

The factors of a polynomial are the polynomials that multiply together to give the original polynomial. For example, $x^{2} - 5 x + 6$ factors to $(x - 2) (x - 3)$ .

Multiplicity

The multiplicity of a root of a polynomial is the number of times that root appears. For example, in $(x - 2)^{3}$ , the root $x = 2$ has a multiplicity of 3.

Odd Multiplicity

If a polynomial has a root with an odd multiplicity, the graph of the polynomial will cross the x-axis at that root.

Even Multiplicity

If a polynomial has a root with an even multiplicity, the graph of the polynomial will touch the x-axis at that root but not cross it.

Turning Point

The turning points of a polynomial are the points where the graph changes direction. A polynomial of degree $n$ can have at most $n - 1$ turning points.

End Behavior of Polynomial

The end behavior of a polynomial describes how the graph behaves as $x$ approaches positive or negative infinity. For example, the end behavior of $f (x) = x^{3}$ is that as $x \to \infty$ , $y \to \infty$ and as $x \to - \infty$ , $y \to - \infty$ .