like terms

Examples of like terms in the following topics:

• Adding and Subtracting Polynomials

  • For example, $4x^3$ and $x^3$are like terms; $21$ and $82$ 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 $x^2$, 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 ($-4x$ and $3x$) 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 $P(x)$ is multiplied exactly once with every term of $Q(x)$.
  • Notice that since the highest degree term of $P(x)$ is multiplied with the highest degree term of $Q(x)$ 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 $M_1(x)$ and end with the terms containing $M_n(x)$.
  • 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 $a_1$, so that the second term is $a_1+d$, the third is $a_1+2d$, and so on.
  • Then our sum looks like:
  • Next, we think of each term as being written in terms of the last term, $a_n$.
  • Then the last term is $a_n$, the term before the last is $a_n-d$, the term before that is $a_n-2d$, 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 $a$.
  • 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 $t_1$.
  • The second term goes up by $d$, and so it is $t_1+d$.
  • The first term is $t_1$; the second term is $r$ times that, or $t_1r$; the third term is $r$ times that, or $t_1r^2$; 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 $t_1+t_n$ .
  • Ditto for the second and next-to-last terms, and so on.
  • Well, there are n terms, so there are $\frac n2$ pairs.