1. Algebra 1
Section 13.4

2. Adding and Subtracting
To add or subtract rational expressions with different denominators, you must first find a common denominator.
The least common denominator (LCD) is the least common multiple (LCM) of the denominators.

3. Adding and Subtracting
To find the LCM of two expressions, factor each expression completely and then find the product of the highest power of each factor.

4. Example 1 Find the LCM of x4 – 3x3 – 10x2 and 3x3 – 30x2 + 75x.

5. Example 2 3 2a 5 8 + = LCD = 23 • a = 8a 4(3) 4(2a) 5a 8a + = 12 8a 5a=
LCD = 23 • a
= 8a
4(3)
4(2a) 5a 8a = 12 8a 5a =
5a + 12 8a

6. Example 3 2x – 5 3x x – 1 2x 5x + 6 12x2 + + LCD = 22 • 3 • x2 = 12x2
LCD = 22 • 3 • x2
= 12x2
4x(2x – 5)
4x(3x)
6x(x – 1)
6x(2x) +
5x + 6
12x2 +

7. Example 3 4x(2x – 5) 4x(3x) 6x(x – 1) 6x(2x) + 5x + 6 12x2 + 8x2 – 20x

8. Example 4 x – 3 x + 1 x – 4 x + 2 – LCD = (x + 1)(x + 2)

9. Example 4 x – 3 x + 1 x – 4 x + 2 – (x – 3)(x + 2) (x + 1)(x + 2)

10. Example 4 x2 – x – 6 (x + 1)(x + 2) x2 – 3x – 4 (x + 2)(x + 1) –

11. Example 5 b a2 + ab a ab + b2 – b a(a + b) a b(a + b) –
LCD = ab(a + b) bb
ab(a + b) aa
ab(a + b) –

12. Example 5 bb ab(a + b) b2 ab(a + b) aa ab(a + b) – a2 ab(a + b) –=

13. Adding and Subtracting
Find the LCD of the rational expressions.
Make equivalent rational expressions having the LCD.

14. Adding and Subtracting
Combine the numerators and place the result over the common denominator.
Simplify the resulting rational expression.

15. Example 6 x + 1 2x – 4 2x – x2 + x + 1 2(x – 2) x(2 – x) + x + 1

16. Example 6 x + 1 2(x – 2) x(x – 2) – LCD = 2x(x – 2) (x + 1)x 2(x – 2)x

17. Example 6 x2 + x 2x(x – 2) 2x + 2 – x2 – x – 2 2x(x – 2)=

18. Homework: pp