5.2 Power Functions and Polynomial Functions

Power Functions

Characteristics of power functions

Power functions take the form $f(x) = kx^n$, where $k$ is a nonzero constant and $n$ is a real number. Every polynomial term is a power function, so understanding these gives you a foundation for working with polynomials overall.

The type of function you get depends on what $n$ is:

• Positive integer $n$: produces polynomial-type behavior (quadratic when $n = 2$, cubic when $n = 3$, etc.)
• Negative integer $n$: produces rational-type behavior, like $x^{-1} = \frac{1}{x}$ or $x^{-2} = \frac{1}{x^2}$
• Fractional $n$: produces root functions, like $x^{1/2} = \sqrt{x}$ or $x^{1/3} = \sqrt[3]{x}$

A few properties worth knowing:

• The graph always passes through $(1, k)$, since $f(1) = k \cdot 1^n = k$ for any $n$.
• Domain is all real numbers when $n$ is a positive integer. When $n$ is negative, $x = 0$ must be excluded (division by zero). When $n$ is a fraction with an even denominator (like $\frac{1}{2}$), the domain is restricted to $x \geq 0$.
• Range depends on both $n$ and the sign of $k$:
  • Even positive integer $n$ with $k > 0$: range is $[0, \infty)$
  • Even positive integer $n$ with $k < 0$: range is $(-\infty, 0]$
  • Odd positive integer $n$: range is $(-\infty, \infty)$ regardless of the sign of $k$
• Symmetry applies when $n$ is a positive integer:
  • Even $n$: symmetric about the y-axis (even function, since $f(-x) = f(x)$)
  • Odd $n$: symmetric about the origin (odd function, since $f(-x) = -f(x)$)

End behavior of power functions

End behavior describes what happens to $f(x)$ as $x$ heads toward $+\infty$ or $-\infty$. For power functions with positive integer exponents, two things control end behavior: whether $n$ is even or odd, and whether $k$ is positive or negative.

1. Even $n$, positive $k$: Both ends rise. As $x \to +\infty$, $f(x) \to +\infty$, and as $x \to -\infty$, $f(x) \to +\infty$.
2. Even $n$, negative $k$: Both ends fall. As $x \to \pm\infty$, $f(x) \to -\infty$.
3. Odd $n$, positive $k$: Opposite ends. As $x \to +\infty$, $f(x) \to +\infty$, and as $x \to -\infty$, $f(x) \to -\infty$.
4. Odd $n$, negative $k$: Opposite ends, flipped. As $x \to +\infty$, $f(x) \to -\infty$, and as $x \to -\infty$, $f(x) \to +\infty$.

This same logic applies to full polynomials: the end behavior of any polynomial is determined entirely by its leading term (the term with the highest degree). So $f(x) = 3x^4 - 7x^2 + 1$ behaves like $3x^4$ at the extremes, meaning both ends rise.

Polynomial Functions

Classification of polynomial functions

A polynomial function has the general form:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

where $n$ is a non-negative integer and $a_n \neq 0$. The key vocabulary:

• Degree: the highest power of $x$ in the polynomial. This single number tells you a lot about the function's shape and behavior.
• Leading coefficient: the coefficient $a_n$ attached to that highest-degree term.
• Leading term: the product $a_n x^n$, which dominates the function's behavior for large $|x|$.

Polynomials are classified by degree:

Manipulation of polynomials

Evaluating a polynomial means plugging in a specific value for $x$ and simplifying. For example, if $f(x) = 2x^3 - x + 1$, then $f(2) = 2(8) - 2 + 1 = 15$.

Polynomial arithmetic follows the same rules you'd expect:

• Addition/Subtraction: Combine like terms (terms with the same power of $x$).
• Multiplication: Distribute each term in one polynomial across every term in the other, then combine like terms.

Factoring is the reverse of multiplication. You're breaking a polynomial into a product of simpler polynomials. The main strategies, roughly in the order you should try them:

1. Factor out the GCF (greatest common factor) first. For $6x^3 + 9x^2$, factor out $3x^2$ to get $3x^2(2x + 3)$.

2. Look for special patterns:

   • Difference of squares: $a^2 - b^2 = (a + b)(a - b)$
   • Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
   • Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

3. Factor by grouping when you have four terms. Group them in pairs, factor each pair, then factor out the common binomial.

4. Trial and error or the AC method for trinomials that don't fit a special pattern.

Finding roots (zeros) means solving $f(x) = 0$. A few tools help here:

• The Fundamental Theorem of Algebra guarantees that a degree-$n$ polynomial has exactly $n$ roots in the complex numbers (counting repeated roots).
• The Rational Root Theorem narrows down possible rational roots: any rational root $\frac{p}{q}$ must have $p$ as a factor of the constant term $a_0$ and $q$ as a factor of the leading coefficient $a_n$.
• Synthetic division or polynomial long division lets you divide out a known root to reduce the polynomial's degree, making it easier to find the remaining roots.

Additional polynomial concepts

A few more ideas that connect to this section:

• Turning points: A degree-$n$ polynomial can have at most $n - 1$ turning points (local maxima or minima). So a cubic can turn at most twice, a quartic at most three times.
• Complex roots: When a polynomial with real coefficients has complex roots, they always come in conjugate pairs. If $2 + 3i$ is a root, then $2 - 3i$ is also a root.
• Polynomial inequalities: To solve something like $x^2 - 4 > 0$, first find the zeros ($x = -2$ and $x = 2$), then test intervals between and beyond those zeros to determine where the polynomial is positive or negative.