like terms

(noun)

Entities that involve the same variables raised to the same exponents.

Examples of like terms in the following topics:

  • Adding and Subtracting Polynomials

    • For example, 4x34x^34x​3​​ and x3x^3x​3​​are like terms; 212121 and 828282 are also like terms.
    • When adding polynomials, the commutative property allows us to rearrange the terms to group like terms together.
    • For example, one polynomial may have the term x2x^2x​2​​, while the other polynomial has no like term.
    • If any term does not have a like term in the other polynomial, it does not need to be combined with any other term.
    • Start by grouping like terms.
  • Adding and Subtracting Algebraic Expressions

    • Terms are called like terms if they involve the same variables and exponents.
    • All constants are also like terms.
    • Likewise, the following are examples of like terms:
    • Now group these like terms together:
    • Now group these like terms together:
  • Sums, Differences, Products, and Quotients

    • In adding equations, it is important to collect like terms to simplify the expression.
    • "Like terms" are those that have the same kind of variable.
    • We then collect like terms.
    • In this case, "x" and "2x" are like terms, as are "5" and "-3. " The result is:
    • It is important to remember to only add together like terms.
  • Multiplying Algebraic Expressions

    • (Note that multiplying monomials is not the same as adding algebraic expressions—monomials do not have to involve "like terms" in order to be combined together through multiplication.)
    • 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)
    • Additionally, remember to simplify the resulting polynomial if possible by combining like terms.
    • Notice that two of these terms are like terms (−4x-4x−4x and 3x3x3x) and can therefore be added together to simplify the expression further:
  • Simplifying Algebraic Expressions

    • A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators.
    • Added terms are simplified using coefficients.
    • Multiplied terms are simplified using exponents.
    • Like terms are added together.
    • Manipulate algebraic expressions by combining like terms and using the distributive property so that they are simplified
  • Multiplying Polynomials

    • To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.
    • and we see that this equals the sum of the products of the terms, where every term of P(x)P(x)P(x) is multiplied exactly once with every term of Q(x)Q(x)Q(x).
    • Notice that since the highest degree term of P(x)P(x)P(x) is multiplied with the highest degree term of Q(x)Q(x)Q(x) we have that the degree of the product equals the sum of the degrees, since
    • For convenience, we will use the commutative property of addition to write the expression so that we start with the terms containing M1(x)M_1(x)M​1​​(x) and end with the terms containing Mn(x)M_n(x)M​n​​(x).
    • This method is commonly called the FOIL method, where we multiply the First, Outside, Inside, and Last pairs in the expression, and then add the products of like terms together.
  • Summing Terms in an Arithmetic Sequence

    • First we think of it as the sum of terms that are written in terms of a1a_1a​1​​, so that the second term is a1+da_1+da​1​​+d, the third is a1+2da_1+2da​1​​+2d, and so on.
    • Then our sum looks like:
    • Next, we think of each term as being written in terms of the last term, ana_na​n​​.
    • Then the last term is ana_na​n​​, the term before the last is an−da_n-da​n​​−d, the term before that is an−2da_n-2da​n​​−2d, and so on.
    • An infinite arithmetic series is exactly what it sounds like: an infinite series whose terms are in an arithmetic sequence.
  • The General Term of a Sequence

    • Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence.
    • Then the sequence looks like:
    • The difference between each term and the term after it is aaa.
    • Then the sequence would look like:
    • If we start at the second term, and subtract the previous term from each term in the sequence, we can get a new sequence made up of the differences between terms.
  • Introduction to Sequences

    • Like a set, it contains members (also called elements or terms).
    • This is more useful, because it means you can find (for instance) the 20th term without finding all of the other terms in between.
    • The first term is always t1t_1t​1​​.
    • The second term goes up by ddd, and so it is t1+dt_1+dt​1​​+d.
    • The first term is t1t_1t​1​​; the second term is rrr times that, or t1rt_1rt​1​​r; the third term is rrr times that, or t1r2t_1r^2t​1​​r​2​​; and so on.
  • Sums and Series

    • A series is merely the sum of the terms of a series.
    • While this trick may not save much time with a 6-item series like the one above, it can be very useful if adding up longer series.
    • If you add the first and last terms, you get t1+tnt_1+t_nt​1​​+t​n​​ .
    • Ditto for the second and next-to-last terms, and so on.
    • Well, there are n terms, so there are n2\frac n2​2​​n​​ pairs.
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.