Algebra 1 Section 13.4

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.

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.

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

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

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 +

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

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

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

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

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) –

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

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

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

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

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

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

Homework: pp. 550-551