SECTION 5.4 Adding and Subtracting Polynomials; Graphing Simple Polynomials^321
CAUTION In Example 2(c),we cannot combine and 5m, because the
exponents on the variables are different. Unlike terms have different variables or
different exponents on the same variables.
OBJECTIVE 3 Know the vocabulary for polynomials. A polynomial in xis
a term or the sum of a finite number of terms of the form for any real number a
and any whole number n. For example,
is a polynomial in x. This polynomial is written in descending powersof the variable,
since the exponents on xdecrease from left to right. By contrast,
,or Not a polynomial
is not a polynomial in x. A variable appears in a denominator or to a negative power.
NOTEWe can define polynomialusing any variable and not just x, as in Example 2(c).
Polynomials may have terms with more than one variable, as in Example 2(d).
The degree of a termis the sum of the exponents on the variables. The degree of
a polynomialis the greatest degree of any nonzero term of the polynomial. The table
gives several examples.
2 x^3 - x^2 + 2 x^3 - x^2 + 4 x-^1 ,
4
x
16 x^8 - 7 x^6 + 5 x^4 - 3 x^2 + 4
axn,
16 m^2
Three types of polynomials are common and are given special names. A polyno-
mial with only one term is called a monomial.(Monomeans “one,” as in monorail.)
MonomialsA polynomial with exactly two terms is called a binomial.(Bi- means “two,” as in
bicycle.)
BinomialsA polynomial with exactly three terms is called a trinomial.(Tri- means “three,” as
in triangle.)
TrinomialsClassifying PolynomialsFor each polynomial, first simplify, if possible. Then give the degree and tell whether
the polynomial is a monomial,a binomial,a trinomial,or none of these.
(a) The polynomial cannot be simplified. It is a binomial of degree 3.
(b)
Add like terms: which is a monomial of degree 2.
NOW TRY4 xy- 5 xy+ 2 xy=xy,
4 xy- 5 xy+ 2 xy
2 x^3 + 5
EXAMPLE 3
9 m^3 - 4 m^2 +6,
19
3
y^2 +
8
3
y+ 5, and - 3 m^5 - 9 m^2 + 2
- 9 x^4 + 9 x^3 , 8 m^2 + 6 m, and 3 m^5 - 9 m^2
9 m, - 6 y^5 , a^2 , and 6
NOW TRY
EXERCISE 3
Simplify, give the degree, and
tell whether the simplified
polynomial is a monomial,a
binomial,a trinomial,or none
of these.
x^2 + 4 x- 2 x- 8Term Degree Polynomial Degree
44
or 1 1
or 0 3
2 x^2 y,or 6 2 x^2 y^12 + 1 = 3 x^5 + 3 x^6- 7, - 7 x^0 x^2 y+xy- 5 y 2
5 x, 5 x^15 x+ 73 x 4 3 x^4 - 5 x^2 + 6Polynomial in x
(The 4 can be written as 4 x^0 .)NOW TRY ANSWER
- x^2 + 2 x-8;degree 2; trinomial