Polynomial Functions: End Behavior and Extrema

• These notes summarize key concepts related to polynomial functions, including end behavior, extrema, real zeros, synthetic substitution, and Descartes' Rule of Signs.

End Behavior of Polynomial Functions

• Refers to how the function behaves as x approaches positive or negative infinity.
• Determined by the leading term of the polynomial (the term with the highest degree).
• If the leading coefficient is positive and the degree is even, the end behavior is up on both sides.
• If the leading coefficient is positive and the degree is odd, the end behavior is down on the left and up on the right.
• If the leading coefficient is negative and the degree is even, the end behavior is down on both sides.
• If the leading coefficient is negative and the degree is odd, the end behavior is up on the left and down on the right.

Extrema of Polynomial Functions

• Extrema refer to the maximum and minimum values of the function.
• Relative (or local) extrema are the maximum or minimum values within a specific interval.
• Absolute extrema are the overall maximum or minimum values of the function.
• Maxima is the point at which a function reaches a relative or absolute highest value.
• Minima is the point at which a function reaches a relative or absolute lowest value.
• These can be found using a graphing calculator.

Real Zeros of Polynomial Functions

• Real zeros are the x-values where the function intersects the x-axis (i.e., where f(x) = 0).
• The number of real zeros can be determined graphically or algebraically.

Increasing and Decreasing Behavior

• Describes the intervals where the function's value is either increasing or decreasing as x increases.
• Increasing behavior means as x increases, y increases.
• Decreasing behavior means as x increases, y decreases.
• These intervals are defined by the x-values of the extrema.

Synthetic Substitution (Remainder Theorem)

• A method for evaluating a polynomial function at a specific value of x.
• It is an alternative to direct substitution.
• Also used to find factors of a polynomial.
• If f(c) = 0, then (x - c) is a factor of the polynomial.

Factoring Polynomials

• Breaking down a polynomial into simpler expressions (factors) that, when multiplied together, give the original polynomial.
• Techniques include factoring out common factors, difference of squares, sum/difference of cubes, and grouping.

Solving Polynomial Equations

• Finding the values of x that make the polynomial equal to zero (also known as roots or zeros).
• Factoring is a common method for solving polynomial equations.
• The number of solutions (real and complex) is equal to the degree of the polynomial.

Descartes' Rule of Signs

• A rule that helps determine the possible number of positive and negative real zeros of a polynomial function.
• The number of positive real zeros is either equal to the number of sign changes in f(x) or less than that by an even number.
• The number of negative real zeros is either equal to the number of sign changes in f(-x) or less than that by an even number.

Building Polynomial Functions from Zeros

• Given the zeros of a polynomial function, one can construct the polynomial by multiplying factors of the form (x - zero).
• For complex zeros, they occur in conjugate pairs (a + bi and a - bi).

Nature of Roots

• Types of roots include real rational, real irrational, and imaginary.
• Real rational roots can be expressed as a ratio of two integers.
• Real irrational roots are real numbers that cannot be expressed as a ratio of two integers (e.g., √2).
• Imaginary roots are complex numbers with a non-zero imaginary part (e.g., a + bi, where b ≠ 0).