Algebraic Functions#
Functions that involve algebraic expressions, such as polynomials and rational functions.
Polynomials#
A polynomial is an expression of the form:
$$P(x) = a_0 + a_1x + a_2x^2 + \ldots + a_{n-1}x^{n-1} + a_nx^n$$
Where:
- \( a_0, a_1, a_2, \ldots, a_{n-1}, a_{n} \) are real numbers (coefficients).
- \( a_n \neq 0 \) (the leading coefficient must be non-zero).
- \( n \) is a non-negative integer. \(n \in {0, 1, 2, \ldots} \) (the degree of the polynomial).
Degree: The highest power of \(x\) in the polynomial.
- Domain: All real numbers [\( x \in \mathbb{R} \) / \(x \in (-\infty, \infty)\)]
Domain: Set of possible values of \(x\) for which the polynomial \(P(x)\) is defined.
Properties of Polynomials#
A polynomial may have any number of terms, but finite. Example:
- The polynomial \(P(x) = 3x^4 - 5x^2 + 2\) has 3 terms, while \(P(x) = 7\) has only 1 term.
But the function \(f(x) = e^x\) has infinitely many terms in its expansion(Taylor series), so it
is not a polynomial.
Another reason why a function with infinitely many terms (like \(e^x\)) is not a polynomial is that in an infinite series we cannot find the degree of the function, which is a key characteristic of polynomials.
- The polynomial \(P(x) = 3x^4 - 5x^2 + 2\) has 3 terms, while \(P(x) = 7\) has only 1 term.
But the function \(f(x) = e^x\) has infinitely many terms in its expansion(Taylor series), so it
is not a polynomial.
The definition of the polynomial function \(P(x)\) should be the same for all values of \(x\) in the domain.
Example:
- The definition of the polynomial function \(P(x) = 4x^3 - 2x + 5\) is the same for all
values of \(x\) in the domain. But the definition of the function \(f(x) = |x|\) is
different for \(x \geq 0\) and \(x < 0\), so it is not a polynomial.
\(f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \)
- The definition of the polynomial function \(P(x) = 4x^3 - 2x + 5\) is the same for all
values of \(x\) in the domain. But the definition of the function \(f(x) = |x|\) is
different for \(x \geq 0\) and \(x < 0\), so it is not a polynomial.
Not all algebraic functions are polynomials#
These are not considered polynomials:
Rational functions: A rational function is a ratio of two polynomials:
$$ f(x) = \frac{P(x)}{Q(x)} $$
where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \neq 0 \). Example:
$$ f(x) = \frac{x^2 + 1}{x - 3} $$
This is not a polynomial because there is a variable in the denominator.
Irrational functions: An irrational function is a function that contains a variable under a root. Example:
$$ f(x) = \sqrt{x^2 + 1} $$
Negative exponents: A function that contains a variable with a negative exponent. Example:
$$ f(x) = x^{-2} + 3 $$
Transcendental functions: Functions that are not algebraic, such as exponential, logarithmic, and trigonometric functions. Example:
$$ f(x) = e^x + \sin(x) $$
Note: Some of these functions can be expressed as infinite series, that is another reason why they are not polynomials.
Piecewise functions: Functions that are defined by different expressions for different intervals of the domain. Examples:
$$ f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 2x + 1 & \text{if } x \geq 0 \end{cases} ; \ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} $$
Types of Polynomials#
- Monomial: A polynomial with only one term. Example: \(P(x) = 5x^3\)
- Binomial: A polynomial with two terms. Example: \(P(x) = 3x^2 - 4\)
- Trinomial: A polynomial with three terms. Example: \(P(x) = 2x^3 + 5x - 1\)
Roots of Polynomials (Quadratic and Cubic Equations)#
The roots of a polynomials are the values of \(x\) for which the polynomial equals zero. So, the points where the graph of the polynomial intersects the x-axis. These are also called the zeros, zero crossings, or solutions of the polynomial equation \(P(x) = 0\).
- For a function \(f(x) = 0\):
where \(\alpha , \beta , \gamma\) are the real roots of the polynomial.
$$f(\alpha) = f(\beta) = f(\gamma) = 0$$ - Complex roots: If a polynomial has complex roots, they will not be visible on the graph as they do not
intersect the x-axis. Complex roots come in conjugate pairs, meaning if \(a + bi\) is a root,
then \(a - bi\) is also a root. Example: The polynomial \(P(x) = x^2 + 4\) has no real roots,
but it has two complex roots: \(2i\) and \(-2i\).
Note: Here, we say it has no zero crossings, but it does have roots (complex roots).