The distributive property can also be applied.can also be written- 2 x- 3
2
.
- 12 x+ 32
2
SECTION 7.1 The Fundamental Property of Rational Expressions 425CAUTION is notan equivalent form of The sign preceding 3
in the numerator of should be rather than. Be careful to apply the
distributive property correctly.
2 x 2 + (^3) - +
12 x+ 32
2.
2 x+ 3
2
Multiply eachterm
in the numerator by -1.Writing Equivalent Forms of a Rational Expression
Write four equivalent forms of the rational expression.If we apply the negative sign to the numerator, we obtain these equivalent forms.1 and, by the distributive property, 2If we apply the negative sign to the denominator, we obtain two more forms.3 or, distributing once again, 43 x+ 2- x+ 6
3 x+ 2- 1 x- 62
- 3 x- 2
x- 6
- 3 x- 2
- 13 x+ 22
x- 6
-
3 x+ 2
x- 6EXAMPLE 7NOW TRYCAUTION Recall that Thus, in Example 7,it would be incorrect to
distribute the negative sign in to boththe numerator andthe denominator.
(Doing this would actually lead to the oppositeof the original expression.)-
3 x+ 2
x- 6-^56 Z --^56.
In Section 5.7,we used long division to find the quotient of two polynomials such as
, as shown on the left. The quotient is. We get
the same quotient by expressing the division problem as a rational expression (frac-
tion) and writing this rational expression in lowest terms, as shown on the right.12 x^2 + 5 x- 122 , 12 x- 32 x+ 4CONNECTIONS
0
8 x- 128 x- 122 x^2 - 3 x2 x- 3 2 x^2 + 5 x- 12x+ 4Factor.=x+ 4 Fundamental property=
12 x- 321 x+ 42
2 x- 32 x^2 + 5 x- 12
2 x- 3For Discussion or Writing
What kind of division problem has a quotient that cannot be found by writing a frac-
tion in lowest terms? Try using rational expressions to solve each division problem.
Then use long division and compare.- 13 x^2 + 11 x+ 82 , 1 x+ 22 2. 1 x^3 - 82 , 1 x^2 + 2 x+ 42
NOW TRY
EXERCISE 7
Write four equivalent forms
of the rational expression.
- 4 k- 9
k+ 3
NOW TRY ANSWER
7.
4 k- 9
- k- 3
4 k- 9- 1 k+ 32 ,
- 4 k+ 9
k+ 3 , - 14 k- 92
k+ 3 ,