Algebra 1 Section 13.1

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

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.

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.

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

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

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

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.

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

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

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

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 =

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.

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) =

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

Example 6 Acircle = πr2 Asquare = s2 = (2r)2 = 4r2 Acircle Asquare = ≈ 0.785 or 78.5%

Homework: pp. 535-537