binomial

(noun)

A polynomial with two terms.

Related Terms

  • binomial coefficient
  • integer
  • square
  • Monomials
  • Binomials
  • Multiplication of Two Monomials
  • Binomial
  • monomial
  • polynomial
  • trinomial

(noun)

A polynomial consisting of two terms, or monomials, separated by addition or subtraction symbols.

Related Terms

  • binomial coefficient
  • integer
  • square
  • Monomials
  • Binomials
  • Multiplication of Two Monomials
  • Binomial
  • monomial
  • polynomial
  • trinomial

(noun)

A polynomial consisting of two terms, or monomials, separated by an addition or subtraction symbol.

Related Terms

  • binomial coefficient
  • integer
  • square
  • Monomials
  • Binomials
  • Multiplication of Two Monomials
  • Binomial
  • monomial
  • polynomial
  • trinomial

Examples of binomial in the following topics:

  • Binomial Expansions and Pascal's Triangle

    • The binomial theorem, which uses Pascal's triangles to determine coefficients, describes the algebraic expansion of powers of a binomial.
    • The binomial theorem is an algebraic method of expanding a binomial expression.
    • This formula is referred to as the Binomial Formula.
    • Applying these numbers to the binomial expansion, we have:
    • Use the Binomial Formula and Pascal's Triangle to expand a binomial raised to a power and find the coefficients of a binomial expansion
  • Complex Numbers and the Binomial Theorem

    • Powers of complex numbers can be computed with the the help of the binomial theorem.
    • Recall the binomial theorem, which tells how to compute powers of a binomial like $x+y$.
    • Using the binomial theorem directly, this can be written as:
    • Recall that the binomial coefficients (from the 5th row of Pascal's triangle) are $1, 5, 10, 10, 5, \text{and}\, 1.$ Using the binomial theorem directly, we have:
    • Connect complex numbers raised to a power to the binomial theorem
  • Multiplying Algebraic Expressions

    • Multiplying two binomials is less straightforward; however, there is a method that makes the process fairly convenient.
    • "FOIL" is a mnemonic for the standard method of multiplying two binomials (hence the method is often referred to as the FOIL method).
    • Outer (the "outside" terms are multiplied—i.e., the first term of the first binomial with the second term of the second)
    • Inner (the "inside" terms are multiplied—i.e., the second term of the first binomial with the first term of the second)
    • Remember that any negative sign on a term in a binomial should also be included in the multiplication of that term.
  • Binomial Expansion and Factorial Notation

    • The binomial theorem describes the algebraic expansion of powers of a binomial.
    • Recall that the binomial theorem is an algebraic method of expanding a binomial that is raised to a certain power, such as $(4x+y)^7$.
    • The coefficients that appear in the binomial expansion are called binomial coefficients.
    • Example: Use the binomial formula to find the expansion of $(x+y)^4$
    • Use factorial notation to find the coefficients of a binomial expansion
  • Total Number of Subsets

    • The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.
    • The binomial theorem describes the algebraic expansion of powers of a binomial.
    • The coefficient a in the term of $ax^by^c$ is known as the binomial coefficient $n^b$ or $n^c$ (the two have the same value).
    • Employ the Binomial Theorem to find the total number of subsets that can be made from n elements
  • Sums, Differences, Products, and Quotients

    • A monomial equations has one term; a binomial has two terms; a trinomial has three terms.
    • Multiplying binomials and trinomials is more complicated, and follows the FOIL method.
    • FOIL is a mnemonic for the standard method of multiplying two binomials; the method may be referred to as the FOIL method.
    • Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second)
    • Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second)
  • Finding a Specific Term

    • The rth term of the binomial expansion can be found with the equation: ${ \begin{pmatrix} n \\ r-1 \end{pmatrix} }{ a }^{ n-(r-1) }{ b }^{ r-1 }$.
    • You might multiply each binomial out to identify the coefficients, or you might use Pascal's triangle.
    • Let's go through a few expansions of binomials, in order to consider any patterns that are present in the terms.
  • Factoring Perfect Square Trinomials

    • When a trinomial is a perfect square, it can be factored into two equal binomials.
    • Note that if a binomial of the form $a+b$ is squared, the result has the following form: $(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2.$ So both the first and last term are squares, and the middle term has factors of $2, $ $a$, and $b,$ where the latter are the square roots of the first and last term respectively.
  • Factoring a Difference of Squares

    • When we multiply together the two binomials $(x-y)$ and $(x+y)$, we obtain the product $x^2-y^2.$ Usually when we multiply two binomials we obtain a trinomial, but in this case, when we FOIL, the outer and inner terms cancel.
  • Combinations

    • The number of $k$-combinations, or $\begin{pmatrix} S \\ k \end{pmatrix}$, is also known as the binomial coefficient, because it occurs as a coefficient in the binomial formula.
    • The binomial coefficient is the coefficient of the $x^k$ term in the polynomial expansion of $(1+x)^n$. $$
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.