Examples of term in the following topics:
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- 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.
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- Terms are called like terms if they involve the same variables and exponents.
- All constants are also like terms.
- Note that terms that share a variable but not an exponent are not like terms.
- Likewise, terms that share an exponent but have different variables are not like terms.
- When an expression contains more than two terms, it may be
helpful to rearrange the terms so that like terms are together.
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- For instance, in the equation y = x + 5, there are two terms, while in the equation y = 2x2, there is only one term.
- We then collect like terms.
- A monomial equations has one term; a binomial has two terms; a trinomial has three terms.
- 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)
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- Any negative sign on a term should be included in the multiplication of that term.
- 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.
- Notice that two of these terms are like terms ($-4x$ and $3x$) and can therefore be added together to simplify the expression further:
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- An arithmetic sequence is a sequence of numbers in which the difference between the consecutive terms is constant.
- An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant.
- Note that the first term in the sequence can be thought of as $a_1+0\cdot d,$ the second term can be thought of as $a_1+1\cdot d,$ the third term can be thought of as $a_1+2\cdot d, $and so the following equation gives $a_n$:
- Of course, one can always write out each term until getting the term sought—but if the 50th term is needed, doing so can be cumbersome.
- Calculate the nth term of an arithmetic sequence and describe the properties of arithmetic sequences
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- 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.
- 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.
- We can see that the first term is $a_1 = 3$.
- The difference between the terms is $d = 5$.
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- The number of terms is one more than $n$ (the exponent).
- The sum of the exponents in each term adds up to $n$.
- Then it is easy to find a particular term.
- Because we are looking for the fifth term, we use $r=5$.
- When the power is applied to the terms, the result is:
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- Given terms in a sequence, it is often possible to find a formula for the general term of the sequence, if the formula is a polynomial.
- Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence.
- In fact, the difference between each pair of terms is $2$.
- The difference between each term and the term after it is $a$.
- 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.
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- Adding the first term to the last term: $3 + 17 = 20$
- Adding the second term to the second-to-last term also amounts to a sum of 20.
- We can see that the third term and third-to-last terms have a similar effect.
- There are eight terms in $3+5+7+9+11+13+15+17$, and they add to four 20s, or 80.
- Adding the first and the last, second term and second to last, etc. all yield the same answer.
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- .$ 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.
- For example, if the expression $2x+3$ were squared, we would obtain $(2x+3)(2x+3)=4x^2+12x+9.$ Note that the first term $4x^2$ is the square of $2x$ while the last term $9$ is the square of $3$, while the middle term is twice $2x\cdot3$.
- Suppose you were trying to factor $x^2+8x+16.$ One can see that the first term is the square of $x$ while the last term is the square of $4$.
- Since the middle term is twice $4 \cdot x$, this must be a perfect square trinomial, and we can factor it as:
- Since the first term is $3x$ squared, the last term is one squared, and the middle term is twice $3x\cdot 1$, this is a perfect square, and we can write: