ALGEBRAIC EXPRESSIONS
An algebraic expression is made up of the
signs and symbols of algebra. These symbols
include the Arabic numerals, literal numbers,
the signs of operation, and so forth. Such an
expression represents one number or one quantity. Thus, just as the sum of 4 and 2 is one
quantity, that is, 6, the sum of c and d is one
quantity, that is, c + d. Likewise , ab,
a  b, and so forth, are algebraic expressions each of which represents one quantity or
number.
Longer expressions may be formed by combinations of the various signs of operation and
the other algebraic symbols, but no matter how
complex such expressions are they still represent one number. Thus the algebraic expression
is one number
The arithmetic value of any algebraic expression depends on the values assigned to the
literal numbers. For example, in the expression 2x^{2}  3ay, if x = 3, a = 5, and y = 1, then
we have the following:
2x^{2}  3ay = 2(3)^{2} 3(5)(1)
= 2(9)  15 = 18  15 = 3
Notice that the exponent is an expression
such as 2x^{2} applies only to the x. If it is desired to indicate the square of 2x, rather than
2 times the square of x, then parentheses are
used and the expression becomes (2x)^{2}.
Practice problems. Evaluate the following
algebraic expressions when a = 4, b = 2, c = 3,
x = 7, and y = 5. Remember, the order of operation is multiplication, division, addition, and
subtraction.
Answers:
1. 53
2. 29
3. 19
4. 53
TERMS AND COEFFICIENTS
The terms of an algebraic expression are
the parts of the expression that are connected
by plus and minus signs. In the expression
3abx + cy  k, for example, 3abx, cy, and k are
the terms of the expression.
An expression containing only one term, such
as 3ab, is called a monomial (mono means one).
A binomial contains two terms; for example,
2r + by. A trinomial consists of three terms.
Any expression containing two or more terms
may also be called by the general name, polynomial (poly means many). Usually special
names are not given to polynomials of more than
three times. The expression x3  3x^{2} + 7x + 1
is a polynomial of four terms. The trinomial
x^{2} + 2x + 1 is an example of a polynomial which
has a special name.
Practice problems. Identify each of the following expressions as a monomial, binomial,
trinomial, or polynomial. (Some expressions
may have two names.)
Answers:
1. Monomial
2. Trinomial (also polynomial)
3. Monomial
4. Polynomial
5. Binomial (also polynomial)
6. Binomial (also polynomial)
In general, a COEFFICIENT of a term is
any factor or group of factors of a term by
which the remainder of the term is to be multiplied. Thus in the term 2axy, 2ax is the coefficient of y, 2a is the coefficient of xy,
and 2 is
the coefficient of axy. The word "coefficient"
is usually used in reference to that factor which
is expressed in Arabic numerals. This factor is sometimes called the NUMERICAL COEFFICIENT. The numerical coefficient is customarily written as the first factor of the term.
In 4x, 4 is the numerical coefficient, or simply
the coefficient, of x. Likewise, in 24xy^{2}, 24 is
the coefficient of xy^{2} and in 16(a + b), 16 is the
coefficient of (a + b). When no numerical coefficient is written it is understood to be 1. Thus
in the term xy, the coefficient is 1.
COMBINING TERMS
When arithmetic numbers are connected by
plus and minus signs, they can always be combined into one number. Thus,
Here three numbers are added algebraically
(with due regard for sign) to give one number.
The terms have been combined into one term.
Terms containing literal numbers can be
combined only if their literal parts are the
same. Terms containing literal factors in
which the same letters are raised to the same
power are called like terms. For example, 3y
and 2y are like terms since the literal parts
are the same. Like terms are added by adding
the coefficients of the like parts. Thus, 3y + 2y
= 5y just as 3 bolts + 2 bolts = 5 bolts. Also
3a^{2}b and a^{2}b are like; 3a^{2}b + a^{2}b =
4a^{2}b and
3a^{2}b  a^{2}b = 2a^{2}b. The numbers ay and by
are
like terms with respect to y. Their sum could
be indicated in two ways: ay + by or (a +
b)y .
The latter may be explained by comparing the
terms to denominate numbers. For instance,
a bolts + b bolts = (a + b) bolts.
Like terms are added or subtracted by adding or subtracting the numerical coefficients
and placing the result in front of the literal
factor, as in the following examples:
7x^{2}  5x^{2} = (7
 5)x^{2} = 2x^{2
} ^{
}5b^{2}x 
3ay^{2} 
8b^{2}x + 10ay^{2} = 3b^{2}x + 7ay^{2}
Dissimilar or unlike terms in an algebraic
expression cannot be combined when numerical
values have not been assigned to the literal
factors. For example, 5x^{2} + 3xy  8y^{2} contains three dissimilar terms. This expression
cannot be further simplified by combining terms
through addition or subtraction. The expression may be rearranged as x(3y  5x) 
8y^{2} or
y(3x  8y)  5x^{2}, but such a rearrangement is
not actually a simplification.
Practice problems. Combine like terms in
the following expression:
Answers:
