Pre-Calculus: Polynomials and Their Functions

Welcome back to Professor Baker's Math Class! Today's lesson, from October 11th, 2005, focused on the fascinating world of polynomials. We started by understanding what polynomials are and how they behave, especially at their 'ends' (end behavior). We also honed our factoring skills to uncover the roots (or zeros) of these functions. Let's recap what we covered.

Understanding Polynomials

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. For example:

$$3x^4 - 7x^2 + 2x - 5$$

is a polynomial. Note that terms with negative or fractional exponents (like $x^{-1}$ or $x^{1/2}$) are not allowed in polynomials.

End Behavior

End behavior describes what happens to the $y$-values of a polynomial function, $f(x)$, as $x$ approaches positive or negative infinity. In other words, what direction does the graph point as you go far to the left and far to the right? The leading term of the polynomial (the term with the highest power of $x$) dictates the end behavior. Key things to consider:

  • Degree: Is the degree (highest power) even or odd?
  • Leading Coefficient: Is the leading coefficient positive or negative?

Here's a quick summary:

  • Even Degree, Positive Leading Coefficient: Graph goes up on both ends (as $x \rightarrow \pm \infty$, $f(x) \rightarrow \infty$).
  • Even Degree, Negative Leading Coefficient: Graph goes down on both ends (as $x \rightarrow \pm \infty$, $f(x) \rightarrow -\infty$).
  • Odd Degree, Positive Leading Coefficient: Graph goes down on the left and up on the right (as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow \infty$, $f(x) \rightarrow \infty$).
  • Odd Degree, Negative Leading Coefficient: Graph goes up on the left and down on the right (as $x \rightarrow -\infty$, $f(x) \rightarrow \infty$ and as $x \rightarrow \infty$, $f(x) \rightarrow -\infty$).

Factoring and Finding Roots

Finding the roots (or zeros) of a polynomial means finding the values of $x$ for which $f(x) = 0$. Factoring is a powerful technique to achieve this. By factoring the polynomial into linear factors, we can easily identify the roots. For example:

$$f(x) = x^2 - 5x + 6$$

can be factored as:

$$f(x) = (x - 2)(x - 3)$$

Therefore, the roots are $x = 2$ and $x = 3$. Remember that the roots of a polynomial are also the $x$-intercepts of its graph.

Important Considerations

Don't forget about the Fundamental Theorem of Algebra, which states that a polynomial of degree $n$ has exactly $n$ complex roots (counting multiplicity). This means a degree 3 polynomial will have 3 roots, a degree 4 polynomial will have 4 roots, and so on. Some roots may be real, and some may be complex (involving imaginary numbers).

Practice Makes Perfect!

Keep practicing your factoring skills and pay close attention to the degree and leading coefficient when determining end behavior. Understanding these concepts is crucial for further exploration of polynomial functions. Keep up the great work, and see you in the next class!