polynomial

Examples of polynomial in the following topics:

• Adding and Subtracting Polynomials

  • Note that any two polynomials can be added or subtracted, regardless of the number of terms in each, or the degrees of the polynomials.
  • The resulting polynomial will have the same degree as the polynomial with the higher degree in the problem.
  • For example, one polynomial may have the term $x^2$, while the other polynomial has no like term.
  • Note that the term $5x^3$ in the first polynomial does not have a like term; neither does $7x$ in the second polynomial.
  • Notice that the answer is a polynomial of degree 3; this is also the highest degree of a polynomial in the problem.

• Introduction to Factoring Polynomials

  • A polynomial consists of a sum of monomials.
  • However, sometimes it will be more useful to write a polynomial as a product of other polynomials with smaller degree, for example to study its zeros.
  • is a factorization of a polynomial of degree $3$ into $3$ polynomials of degree $1$.
  • The aim of factoring is to reduce objects to "basic building blocks", such as integers to prime numbers, or polynomials to irreducible polynomials.
  • One way to factor polynomials is factoring by grouping.

• The Fundamental Theorem of Algebra

  • The fundamental theorem of algebra says that every non-constant polynomial in a single variable $z$, so any polynomial of the form
  • For example, the polynomial
  • So since the property is true for all polynomials of degree $0$, it is also true for all polynomials of degree $1$.
  • And since it is true for all polynomials of degree $1$, it is also true for all polynomials of degree $2$.
  • The multiplicities of the complex roots of a nonzero polynomial with complex coefficients add to the degree of said polynomial.

• Polynomial Inequalities

  • The best way to solve a polynomial inequality is to find its zeros.
  • The easiest way to find the zeros of a polynomial is to express it in factored form.
  • Graph of the third-degree polynomial with the equation $y=x^3+2x^2-5x-6$.
  • This polynomial has three roots.
  • Solve for the zeros of a polynomial inequality to find its solution

• Basics of Graphing Polynomial Functions

  • A polynomial function in one real variable can be represented by a graph.
  • Polynomials appear in a wide variety of areas of mathematics and science.
  • A typical graph of a polynomial function of degree 3 is the following:
  • A polynomial of degree 6.
  • A polynomial of degree 5.

• Multiplying Polynomials

  • To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.
  • So for the multiplication of a monomial with a polynomial we get the following procedure:
  • To multiply a polynomial $P(x) = M_1(x) + M_2(x) + \ldots + M_n(x)$ with a polynomial $Q(x) = N_1(x) + N_2(x) + \ldots + N_k(x)$, where both are written as a sum of monomials of distinct degrees, we get
  • Since we made sure that the product of polynomials abides the same laws as if the variables were real numbers, the evaluation of a product of two polynomials in a given point will be the same as the product of the evaluations of the polynomials:
  • So the roots of a product of polynomials are exactly the roots of its factors, i.e.

• Dividing Polynomials

  • Polynomial long division is a method for dividing a polynomial by another polynomial of the same or lower degree.
  • The calculated polynomial is the quotient, and the number left over (−123) is the remainder: $x^3 - 12x^2 - 42 = (x - 3)(x^2 - 9x - 27) - 123$.
  • Polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree.This method is a generalized version of the familiar arithmetic technique called long division.It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.
  • The calculated polynomial is the quotient, and the number left over (−123) is the remainder:

• Polynomial Regression

  • Polynomial regression is a higher order form of linear regression in which the relationship between the independent variable $x$ and the dependent variable $y$ is modeled as an $n$th order polynomial.
  • Polynomial regression models are usually fit using the method of least-squares.
  • More recently, the use of polynomial models has been complemented by other methods, with non-polynomial models having advantages for some classes of problems.
  • Although polynomial regression is technically a special case of multiple linear regression, the interpretation of a fitted polynomial regression model requires a somewhat different perspective.
  • Some of these methods make use of a localized form of classical polynomial regression.

• What Are Polynomials?

  • Polynomials are widely used algebraical objects.
  • The degree of the zero polynomial is defined to be $-\infty$.
  • We have discussed polynomials over $\mathbb{R}$.
  • In this case, we talk about complex polynomials, or polynomials over $\mathbb{C}$.
  • The polynomials over this ring will be polynomials in two variables $x$ and $y$ over $\mathbb{R}$.

• Finding Zeros of Factored Polynomials

  • The factored form of a polynomial can reveal where the function crosses the $x$-axis.
  • A polynomial function may have many, one, or no zeros.
  • This is why factorization is so important: to be able to recognize the zeros of a polynomial quickly.
  • It follows from the fundamental theorem of algebra and a fact called the complex conjugate root theorem, that every polynomial with real coefficients can be factorized into linear polynomials and quadratic polynomials without real roots.
  • Use the factored form of a polynomial to find its zeros