302 CHAPTER 5 Exponents, Polynomials, and Polynomial Functions
OBJECTIVES OBJECTIVE 1 Divide a polynomial by a monomial. Recall that a monomial
is a single term, such as 3, 5m^2 ,or x^2 y^2.
Dividing Polynomials
5.5
1 Divide a polynomial
by a monomial.
2 Divide a polynomial
by a polynomial of
two or more terms.
3 Divide polynomial
functions.Dividing a Polynomial by a MonomialTo divide a polynomial by a monomial, divide each term in the polynomial by the
monomial, and then write each quotient in lowest terms.
Dividing a Polynomial by a MonomialDivide.
(a)
Divide each term by 3.Write in lowest terms.CHECK ✓
DivisorQuotient Original polynomial (Dividend)(b)
Divide each term bySimplify each term.
Use the quotient rule for exponents.CHECK ✓ Divisor Quotient
Original polynomialThe result is not a polynomial. (Why?) The quotient of two polynomials
need not be a polynomial.
(c)
Divide each term by.a NOW TRY
m
an=a=^8 m-n
x
-
9
y
+ 6
= x^2 y^2
8 xy^2
x^2 y^2
-
9 x^2 y
x^2 y^2
+
6 x^2 y^2
x^2 y^2
8 xy^2 - 9 x^2 y+ 6 x^2 y^2
x^2 y^2
m-
95 +
2
m5 m^2 am-^9 * =
5
+
2
m
b = 5 m^3 - 9 m^2 + 10 m
= m-
9
5
+
2
m
= 5 m^2.
5 m^3
5 m^2
-
9 m^2
5 m^2
+
10 m
5 m^2
5 m^3 - 9 m^2 + 10 m
5 m^2
315 x^2 - 4 x+ 22 = 15 x^2 - 12 x+ 6
= 5 x^2 - 4 x+ 2
=
15 x^2
3
-
12 x
3
+
6
3
15 x 2 - 12 x+ 6
3
EXAMPLE 1
⎧⎪⎪⎨⎪⎪⎩ ⎧⎪ ⎪⎨ ⎪⎪ ⎩
NOW TRY
EXERCISE 1
Divide.
81 y^4 - 54 y^3 + 18 y
9 y^3
NOW TRY ANSWER
- 9 y- 6 +
2
y^2
Think:^105 mm 2 =^105 m^1 -^2 = 2 m-^1 =m^2