NOTE If a polynomial in a single variable is written in descending powers of that
variable, the degree of the polynomial will be the degree of the leading term.
280 CHAPTER 5 Exponents, Polynomials, and Polynomial Functions
Classifying PolynomialsIdentify each polynomial as a monomial, binomial, trinomial,or none of these.Also,
give the degree.
(a) This is a trinomial of degree 2.
(b) This is a monomial of degree 5
(c) This is a binomial of degree 14.
(d) Polynomials of four terms or more do not have special
names, so none of theseis the answer that applies here.
This polynomial has degree 4.
p^4 - p^2 - 6 p- 5
7 m^9 + 18 m^14
43 xy^4 Aor 43 x^1 y^4 B^1 because 1+^4 =^52.
- x^2 + 5 x+ 1
EXAMPLE 2
OBJECTIVE 2 Add and subtract polynomials. We use the distributive prop-
erty to simplify polynomials by combining like terms.
Distributive property
Add.Notice that the terms in a polynomial such as cannot be combined.
Only terms containing exactly the same variables to the same powers may be com-
bined. Recall that such terms are called like terms.
4 x+ 5 x^2
=x^3 + 9 x^2 - 1
=x^3 + 14 + 52 x^2 - 1
x^3 + 4 x^2 + 5 x^2 - 1
CAUTION Only like terms can be combined.
Combining Like TermsCombine like terms.
(a)
Distributive property
Add and subtract.(b)
Commutative property
Combine like terms.Since and 7yare unlike terms, no further simplification is possible.
(c)
Commutative property= 14 x^2 y+ 7 xy 2 Combine like terms. NOW TRY
= 5 x^2 y+ 9 x^2 y- 6 x y^2 + 13 x y^2
5 x^2 y- 6 x y^2 + 9 x^2 y+ 13 x y^2
- 3 x
= - 3 x+ 7 y
= 6 x- 9 x+ 5 y+ 2 y
6 x+ 5 y- 9 x+ 2 y
= 2 y^3
= 1 - 5 + 8 - 12 y^3
- 5 y^3 + 8 y^3 - y^3
EXAMPLE 3
Adding PolynomialsTo add two polynomials, combine like terms.
NOW TRY
EXERCISE 2
Identify the polynomial as a
monomial, binomial,
trinomial,or none of these.
Also, give the degree.
- 4 x^3 + 10 x^5 + 7
NOW TRY
EXERCISE 3
Combine like terms.
(a)
(b) -x^2 t+ 4 x^2 t+ 3 xt^2 - 7 xt^2
3 p^3 - 2 q+p^3 - 5 qNOW TRY ANSWERS
- trinomial; 5
- (a) 4 p^3 - 7 q (b) 3 x^2 t- 4 xt^2
NOW TRY