1. Algebra 1
Section 13.1

2. Rational Numbers
Properties that apply to rational numbers can also be applied to rational expressions.

3. Definitions
A rational expression is a ratio of polynomials whose denominator is not zero.
An excluded value is a value of the variable that makes the denominator equal to zero.

4. Rational Expressions x is a rational expression. x + 2
-2 is an excluded value of the rational expression, since when x = -2, the denominator is equal to zero.

5. Example 1a x – 6 x – 9 The denominator cannot equal zero. x – 9 ≠ 0

6. Example 1b 2x2 + 3x + 6 x2 – 8x + 7 x2 – 8x + 7 ≠ 0 (x – 7)(x – 1) ≠ 0
x – 7 ≠ 0 or x – 1 ≠ 0
x ≠ 7, 1

7. Example 1c x + 7 x2 + 3x – 9 x2 + 3x – 9 ≠ 0 -3 ± 3 5
-3 ± 3 5 2
x ≠
-3 ± 32 – 4(1)(-9)
2(1)
≈ 1.85, -4.85

8. Rational Expressions
Some rational expressions have no excluded values.
To simplify a rational expression, factor its numerator and denominator and cancel out any common factors.

9. Example 2b
180m3n
27mn2
9 • 20mm2n
9 • 3mnn
20m2 3n

10. Example 3c
3x + 6 3
3(x + 2) 3
x + 2

11. Rational Expressions Common factors can be canceled.
Common terms cannot be canceled.
x + 2 2
No!

12. Notice that the excluded values are x = ±2.
Example 4
x2 + 4x + 4
x2 – 4
Notice that the excluded values are x = ±2.
(x + 2)(x + 2)
(x – 2)(x + 2)
x + 2
x – 2 =

13. Rational Expressions
Factors such as x – 3 and 3 – x are opposites of each other.
Factoring -1 from one of the factors allows you to cancel a common factor.

14. Notice that the excluded value is x = 3.
Example 5
4x2 + 4x – 48
3 – x
Notice that the excluded value is x = 3.
4(x2 + x – 12)
3 – x
4(x – 3)(x + 4)
-1(x – 3) =

15. Example 5
4(x – 3)(x + 4)
-1(x – 3) =
4(x + 4) -1
= -4(x + 4)
; x ≠ 3

16. Example 6 Acircle = πr2 Asquare = s2 = (2r)2 = 4r2 Acircle Asquare =
≈ or 78.5%

17. Homework: pp