Polynomials

Polynomials ☰

Polynomials are mathematical expressions involving a sum of terms, each term being a product of a constant coefficient and a variable raised to a non-negative integer power. Polynomials are fundamental in mathematics and have a wide range of applications in fields such as algebra, calculus, and number theory.
A polynomial \(P(x)\) in one variable \(x\) is typically written in the form: $$ P(x) = a_n x^n + a_{n-1} x^{n-1} + ⋯ + a_1 x + a_0 $$ Here, \(n\) is a non-negative integer called the degree of the polynomial, and \(a_i\) (for \(i = 0,1,...,n\) ) are the constant coefficients, where \(a_n \neq 0 \) for \( n \ge 1 \). The variable \(x\) is called the indeterminate of the polynomial, and each term \(a_i x^i \) is called a monomial.

Some examples of polynomials:

• \( P(x) = 5x^3 - 2x^2 + 4x-1 \) This is a polynomial of degree 3 (cubic polynomial).
• \( Q(x) = 7x^4 - 3x + 9 \) This is a polynomial of degree 4 (quartic polynomial).
• \( R(x) = 6x - 3 \) This is a polynomial of degree 1 (linear polynomial).

Important properties and concepts related to polynomials include:
Roots or Zeros: A root (or zero) of a polynomial \(P(x)\) is a value of x for which the polynomial evaluates to zero, i.e., \(P(x)=0\). The Fundamental Theorem of Algebra states that a non-constant polynomial of degree n will have exactly n (not necessarily distinct) complex roots, counting multiplicities.

Polynomial addition and subtraction: To add or subtract two polynomials, simply add or subtract their corresponding coefficients.
For example, if $$ P(x)=3x^2 +2x–1 $$ and $$ Q(x)=x^2 –x+4 $$ then $$ P(x)+Q(x)=(3+1) x^2+(2–1)x+ (-1+4)=4x^2+x+3 $$

Polynomial multiplication: To multiply two polynomials, apply the distributive law and combine like terms.
For example, if \( P(x)=2x^2+x–3 \) and \( Q(x)=x–1 \), then $$ P(x) \cdot Q(x)=(2x^2+x–3)(x–1)= 2x^3–2x^2+x^2–x–3x+3=2x^3–x^2–4x+3 $$

Polynomial division: Dividing one polynomial by another involves finding the quotient and remainder. The division algorithm for polynomials states that for any polynomials \( P(x) \) and \( D(x) \), where \(D(x) \neq 0 \), there exist unique polynomials \( Q(x) \) and \( R(x) \) such that $$ P(x)=D(x)Q(x)+R(x) $$ where either \(R(x)=0 \) or the degree of \(R(x)\) is less than the degree of \( D(x) \). This is analogous to the division of integers, where the quotient and remainder are uniquely determined.
Long division and synthetic division are two common methods used to perform polynomial division.

Factoring: Factoring a polynomial involves expressing it as a product of simpler polynomials.
For example, the polynomial \( x^2–5x+6 \) can be factored as \((x–2)(x–3) \). Factoring is an important tool in finding the roots of a polynomial, as a polynomial is equal to zero if and only if one of its factors is equal to zero.

Greatest common divisor (GCD): The GCD of two polynomials is the polynomial of the highest degree that divides both polynomials without leaving a remainder. The Euclidean algorithm can be used to find the GCD of two polynomials, similar to how it is used for integers.

Polynomial functions: A polynomial function is a function defined by a polynomial expression. These functions have many important properties and are widely used in mathematics and science. For example, polynomial functions are continuous and differentiable on their entire domain, which makes them useful in calculus and mathematical modeling.

Interpolation: Polynomials can be used to approximate or interpolate a given set of data points. One common method is Lagrange interpolation, which constructs a polynomial that passes through all the given data points.

Polynomial equations: A polynomial equation is an equation in which one side is a polynomial expression and the other side is either a constant or another polynomial expression. Solving polynomial equations involves finding the values of the variable for which the equation holds true. Techniques for solving polynomial equations include factoring, applying the Rational Root Theorem, and using numerical methods such as the Newton-Raphson method.

Polynomial functions ☰

Polynomial functions are a class of functions that can be represented by a polynomial expression. A polynomial function \(P(x)\) in one variable \(x\) is typically written in the form: $$ P(x)=a_n x^n +a_{n-1} x{n-1} + ⋯ + a_1 x+a_0 $$ Here, \(n\) is a non-negative integer called the degree of the polynomial, and \(a_i\) (for \(i = 0,1,…,n \) ) are the constant coefficients, where \(a_n \neq 0 \) for \( n \ge 1 \). The variable \(x\) is called the indeterminate of the polynomial, and each term \(a_i x^i \) is called a monomial.

Polynomial functions have many important properties that make them useful in various branches of mathematics and science:
Continuity: Polynomial functions are continuous on their entire domain, which is the set of all real numbers. This means that there are no gaps or jumps in the graph of a polynomial function.

Differentiability: Polynomial functions are differentiable everywhere in their domain, meaning they have a derivative at every point. The derivative of a polynomial function is another polynomial function, obtained by differentiating each term with respect to \(x\).
For example, if $$ P(x)=5x^3–2x^2+4x–1 $$ then $$ P' (x)=15x^2–4x+4 $$ Smoothness: As a consequence of being differentiable, polynomial functions are smooth, meaning they do not have any sharp corners or cusps in their graph.

End behavior: The end behavior of a polynomial function is determined by its leading term, which is the term with the highest degree. As \(x\) approaches positive or negative infinity, the leading term dominates the behavior of the function, and the other terms become less significant.
For example, if $$ P(x)=3x^4–5x^2+2x–1 $$ then the end behavior of \(P(x)\) is determined by \(3x^4\), so the graph will grow without bound as \(x\) approaches positive or negative infinity.

Root-finding: Polynomial functions can be used to model and solve problems where the goal is to find the values of x for which the function equals zero. Techniques for finding the roots of a polynomial function include factoring, applying the Rational Root Theorem, and using numerical methods such as the Newton-Raphson method.

Interpolation: Polynomial functions can be used to approximate or interpolate a given set of data points. One common method is Lagrange interpolation, which constructs a polynomial that passes through all the given data points.

Basis for function spaces: Polynomial functions form a basis for various function spaces, such as the space of continuous functions or differentiable functions. This means that any function in these spaces can be approximated arbitrarily closely by a polynomial function. This property is utilized in many areas of mathematics, including approximation theory, numerical analysis, and functional analysis.

Taylor series and approximations: Polynomial functions play a significant role in approximating more complex functions through Taylor series. A Taylor series is an infinite sum of terms that represents a function as a power series of its derivatives around a particular point. If a function is sufficiently smooth, its Taylor series converges to the function in some neighborhood of the point. Truncating the Taylor series after a certain number of terms results in a polynomial that approximates the original function near the point of expansion.

Orthogonal polynomials: There is a special class of polynomial functions called orthogonal polynomials, which have properties that make them useful in various applications, such as solving differential equations, numerical integration, and signal processing. Some well-known families of orthogonal polynomials include Legendre polynomials, Hermite polynomials, and Chebyshev polynomials.

Algebraic properties: Polynomial functions exhibit several algebraic properties, including closure under addition, subtraction, multiplication, and composition. This means that when you perform these operations on polynomial functions, you obtain another polynomial function.
For example, if \(P(x)\) and \(Q(x)\) are polynomial functions, then their sum, difference, product, and composition \( (P \circ Q)(x)=P(Q(x)) \) are also polynomial functions.

Graphs of polynomial functions: The graph of a polynomial function is a curve in the Cartesian plane that represents the set of all points \((x,P(x))\). The shape of the graph depends on the degree of the polynomial and the signs of its coefficients. For example, the graph of a linear polynomial is a straight line, and the graph of a quadratic polynomial is a parabola.

Polynomial regression: In statistics, polynomial functions can be used to fit a curve to a set of data points through a method called polynomial regression. This involves finding a polynomial function of a specified degree that best fits the data, usually by minimizing the sum of squared errors between the observed data points and the predicted values from the polynomial function.

In summary, polynomial functions are a versatile and fundamental class of functions in mathematics and science. Their properties, such as continuity, differentiability, smoothness, and their ability to approximate more complex functions, make them indispensable tools in various applications, ranging from algebra and calculus to numerical analysis, approximation theory, and statistical modeling.

Rational functions ☰

Rational functions are mathematical expressions that represent the ratio of two polynomial functions. In general, a rational function can be written as: $$ R(x) = \frac{P(x)}{Q(x)} $$ where \(P(x)\) and \(Q(x)\) are polynomial functions, and \(Q(x)\) is not equal to zero.
Both \(P(x)\) and \(Q(x)\) can be written as a sum of terms with coefficients and variables raised to non-negative integer powers: $$ \small P(x)= a_n \cdot x^n + a_{n-1} \cdot x^{n-1} + … + a_1 \cdot x+a_0 $$ $$ \small Q(x)=b_m \cdot x^m +b_{m-1} \cdot x^{m-1} + … + b_1 \cdot x+b_0 $$ The domain of a rational function consists of all real numbers \(x\) for which the denominator \(Q(x)\) is not equal to zero. In other words, the domain is the set of all real numbers except for those that make the denominator zero.

Some properties of rational functions include:
Vertical asymptotes: These occur when the denominator \(Q(x)\) equals zero, and the function approaches infinity or negative infinity as \(x\) approaches the value at which the denominator is zero.

Horizontal asymptotes: These occur when the degrees of the numerator and denominator are equal or when the degree of the numerator is less than the degree of the denominator. A horizontal asymptote represents the limit of the function as \(x\) approaches positive or negative infinity.

Holes: These are points that are not in the domain of the rational function due to the cancellation of a factor in both the numerator and the denominator.

Intercepts: To find the \(x\)-intercept(s), set the numerator \(P(x)\) equal to zero and solve for \(x\). To find the \(y\)-intercept, set \(x\) equal to zero and find the corresponding value of \(R(x)\).

End behavior: The end behavior of a rational function is determined by the degrees of the numerator and the denominator, as well as the coefficients of the leading terms in both the numerator and the denominator.

To analyze and graph rational functions, it's helpful to find the intercepts, asymptotes, holes, and end behavior of the function. This information can be used to sketch the general shape of the graph and understand the behavior of the function in different regions of its domain.