polynomial

(noun)

an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as $a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0$. Importantly, because all exponents are positive, it is impossible to divide by x.

Related Terms

  • indeterminate
  • Monomials
  • Binomials
  • Multiplication of Two Monomials
  • Binomial
  • term
  • binomial
  • monomial
  • radicand
  • greatest common divisor
  • remainder
  • factorization
  • rational expression
  • degree
  • coefficient
  • zero
  • commutative
  • expression
  • trinomial

(noun)

an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as $a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0$. Importantly, because all exponents are positive, it is impossible to divide by $x$.

Related Terms

  • indeterminate
  • Monomials
  • Binomials
  • Multiplication of Two Monomials
  • Binomial
  • term
  • binomial
  • monomial
  • radicand
  • greatest common divisor
  • remainder
  • factorization
  • rational expression
  • degree
  • coefficient
  • zero
  • commutative
  • expression
  • trinomial

(noun)

an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as $a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0$. 

Related Terms

  • indeterminate
  • Monomials
  • Binomials
  • Multiplication of Two Monomials
  • Binomial
  • term
  • binomial
  • monomial
  • radicand
  • greatest common divisor
  • remainder
  • factorization
  • rational expression
  • degree
  • coefficient
  • zero
  • commutative
  • expression
  • trinomial

(noun)

An algebraic expression with more than one term.

Related Terms

  • indeterminate
  • Monomials
  • Binomials
  • Multiplication of Two Monomials
  • Binomial
  • term
  • binomial
  • monomial
  • radicand
  • greatest common divisor
  • remainder
  • factorization
  • rational expression
  • degree
  • coefficient
  • zero
  • commutative
  • expression
  • trinomial

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:
  • 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
  • Zeroes of Polynomial Functions With Rational Coefficients

    • Polynomials with rational coefficients should be treated and worked the same as other polynomials.
    • Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a "polynomial over the rationals".
    • Polynomials with rational coefficients can be treated just like any other polynomial, just remember to utilize all the properties of fractions necessary during your operations.
    • Graph of a polynomial with the quadratic equation of $y=\frac{2x^2}{9}+\frac{7x}{3}+6$.
    • Extend the techniques of finding zeros to polynomials with rational coefficients
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