Polynomial and Rational Functions

Definition and Properties of Polynomial Functions

A polynomial function is an expression of the form , where and is a non-negative integer. The degree of the polynomial is the highest power of with a nonzero coefficient.

• Degree: Indicates the highest exponent in the polynomial.

• Continuity: Polynomial functions are continuous (no breaks or holes) and smooth (no sharp corners).

Zeros of Polynomial Functions

The zeros of a polynomial are the values of for which . These can be real or complex numbers.

• Rational Zero Theorem: Provides a method to find possible rational zeros of a polynomial with integer coefficients.

• Factoring: Once zeros are found, the polynomial can be factored as .

• Multiplicity: The number of times a zero occurs. If a zero has even multiplicity, the graph touches the x-axis at that point; if odd, it crosses the axis.

Operations with Complex Numbers

Complex numbers are of the form , where .

• Addition:

• Multiplication:

• Division:

Solving Polynomial Inequalities

To solve inequalities involving polynomials, find the zeros and test intervals between them.

• Set the polynomial equal to zero to find critical points.

• Test values in each interval to determine where the inequality holds.

Rational Functions

A rational function is a function of the form , where and are polynomials and .

• Lowest Terms: Simplify by factoring and canceling common factors.

• Vertical Asymptotes: Occur at zeros of (where the denominator is zero).

• Horizontal Asymptotes: Determined by the degrees of and .

• Graphing: Identify intercepts and asymptotes before sketching the graph.

Solving Rational Inequalities

Similar to polynomial inequalities, but also consider points where the denominator is zero.

• Find zeros of numerator and denominator.

• Test intervals between critical points.

Synthetic Division

Synthetic division is a shortcut for dividing a polynomial by a linear factor .

• Write coefficients in a row.

• Use the value to perform the division.

Finding a Polynomial from Zeros and a Point

If given zeros and a point , construct the polynomial and solve for the leading coefficient using the point.

• Form:

• Plug in the point to solve for .

Exponential and Logarithmic Functions

Composition and Inverses of Functions

The composition of two functions and is . The domain of the composite function is the set of values for which is in the domain of .

• 1-to-1 Function: A function is one-to-one if each output corresponds to exactly one input.

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function is not one-to-one.

• Invertibility: Only one-to-one functions have inverses.

• Inverse Function: If is invertible, satisfies .

Exponential Functions

An exponential function has the form , where is the initial value and is the growth rate.

• Domain: All real numbers.

• Range: Positive real numbers if and .

• Graphing: Exponential functions increase or decrease rapidly, depending on .

• Example:

The Number e

The constant is defined as , and .

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. The general form is , where and .

• Domain:

• Range: All real numbers

• Conversion:

Properties of Logarithms

Solving Logarithmic and Exponential Equations

• Apply logarithm properties to simplify equations.

• Convert between exponential and logarithmic forms as needed.

Applications: Exponential Growth and Decay

Many real-world problems use the model , where is the initial amount, is the growth (or decay) rate, and is time.

• Growth:

• Decay:

• Example: Population growth, radioactive decay

Additional info: The study guide references practice problems and answers in the textbook, which are not included here.