Simplify a rational expression EXAMPLE 1 Simplify a rational expression x2 – 2x – 15 x2 – 9 Simplify : SOLUTION x2 – 2x – 15 x2 – 9 (x +3)(x –5) (x +3)(x –3) = Factor numerator and denominator. (x +3)(x –5) (x +3)(x –3) = Divide out common factor. x – 5 x – 3 = Simplified form ANSWER x – 5 x – 3

Standardized Test Practice EXAMPLE 3 Standardized Test Practice SOLUTION 8x 3y 2x y2 7x4y3 4y 56x7y4 8xy3 = Multiply numerators and denominators. 8 7 x x6 y3 y 8 x y3 = Factor and divide out common factors. 7x6y = Simplified form The correct answer is B. ANSWER

Multiply rational expressions EXAMPLE 4 Multiply rational expressions x2 + x – 20 3x 3x –3x2 x2 + 4x – 5 Multiply: SOLUTION x2 + x – 20 3x 3x –3x2 x2 + 4x – 5 3x(1– x) (x –1)(x +5) = (x + 5)(x – 4) 3x Factor numerators and denominators. 3x(1– x)(x + 5)(x – 4) = (x –1)(x + 5)(3x) Multiply numerators and denominators. 3x(–1)(x – 1)(x + 5)(x – 4) = (x – 1)(x + 5)(3x) Rewrite 1– x as (– 1)(x – 1). 3x(–1)(x – 1)(x + 5)(x – 4) = (x – 1)(x + 5)(3x) Divide out common factors.

Multiply rational expressions EXAMPLE 4 Multiply rational expressions = (–1)(x – 4) Simplify. = –x + 4 Multiply. ANSWER –x + 4

Multiply a rational expression by a polynomial EXAMPLE 5 Multiply a rational expression by a polynomial Multiply: x + 2 x3 – 27 (x2 + 3x + 9) SOLUTION x + 2 x3 – 27 (x2 + 3x + 9) = x + 2 x3 – 27 x2 + 3x + 9 1 Write polynomial as a rational expression. (x + 2)(x2 + 3x + 9) (x – 3)(x2 + 3x + 9) = Factor denominator. (x + 2)(x2 + 3x + 9) (x – 3)(x2 + 3x + 9) = Divide out common factors. = x + 2 x – 3 Simplified form ANSWER x + 2 x – 3

Divide rational expressions EXAMPLE 6 Divide rational expressions Divide : 7x 2x – 10 x2 – 6x x2 – 11x + 30 SOLUTION 7x 2x – 10 x2 – 6x x2 – 11x + 30 7x 2x – 10 x2 – 6x x2 – 11x + 30 = Multiply by reciprocal. 7x 2(x – 5) = (x – 5)(x – 6) x(x – 6) Factor. = 7x(x – 5)(x – 6) 2(x – 5)(x)(x – 6) Divide out common factors. 7 2 = Simplified form ANSWER 7 2

Divide a rational expression by a polynomial EXAMPLE 7 Divide a rational expression by a polynomial Divide : 6x2 + x – 15 4x2 (3x2 + 5x) SOLUTION 6x2 + x – 15 4x2 (3x2 + 5x) 6x2 + x – 15 4x2 3x2 + 5x = 1 Multiply by reciprocal. (3x + 5)(2x – 3) 4x2 = x(3x + 5) 1 Factor. (3x + 5)(2x – 3) = 4x2(x)(3x + 5) Divide out common factors. 2x – 3 4x3 = Simplified form ANSWER 2x – 3 4x3

Add or subtract with like denominators EXAMPLE 1 Add or subtract with like denominators Perform the indicated operation. 7 4x + 3 a. 2x x + 6 – 5 b. SOLUTION 7 4x + 3 a. = 7 + 3 4x 10 4x = 5 2x = Add numerators and simplify result. 2x x + 6 5 – b. x + 6 2x – 5 = Subtract numerators.

EXAMPLE 2 Find a least common multiple (LCM) Find the least common multiple of 4x2 –16 and 6x2 –24x + 24. SOLUTION STEP 1 Factor each polynomial. Write numerical factors as products of primes. 4x2 – 16 = 4(x2 – 4) = (22)(x + 2)(x – 2) 6x2 – 24x + 24 = 6(x2 – 4x + 4) = (2)(3)(x – 2)2

EXAMPLE 2 Find a least common multiple (LCM) STEP 2 Form the LCM by writing each factor to the highest power it occurs in either polynomial. LCM = (22)(3)(x + 2)(x – 2)2 = 12(x + 2)(x – 2)2

Add with unlike denominators EXAMPLE 3 Add with unlike denominators Add: 9x2 7 + x 3x2 + 3x SOLUTION To find the LCD, factor each denominator and write each factor to the highest power it occurs. Note that 9x2 = 32x2 and 3x2 + 3x = 3x(x + 1), so the LCD is 32x2 (x + 1) = 9x2(x 1 1). 7 9x2 x 3x2 + 3x = + 3x(x + 1) Factor second denominator. 7 9x2 x + 1 + 3x(x + 1) x 3x LCD is 9x2(x + 1).

Add with unlike denominators EXAMPLE 3 Add with unlike denominators 7x + 7 9x2(x + 1) 3x2 + = Multiply. 3x2 + 7x + 7 9x2(x + 1) = Add numerators.

Subtract with unlike denominators EXAMPLE 4 Subtract with unlike denominators Subtract: x + 2 2x – 2 –2x –1 x2 – 4x + 3 – SOLUTION x + 2 2x – 2 –2x –1 x2 – 4x + 3 – x + 2 2(x – 1) – 2x – 1 (x – 1)(x – 3) – = Factor denominators. x + 2 2(x – 1) = x – 3 – – 2x – 1 (x – 1)(x – 3) 2 LCD is 2(x  1)(x  3). x2 – x – 6 2(x – 1)(x – 3) – 4x – 2 – = Multiply.

Subtract with unlike denominators EXAMPLE 4 Subtract with unlike denominators x2 – x – 6 – (– 4x – 2) 2(x – 1)(x – 3) = Subtract numerators. x2 + 3x – 4 2(x – 1)(x – 3) = Simplify numerator. = (x –1)(x + 4) 2(x – 1)(x – 3) Factor numerator. Divide out common factor. x + 4 2(x –3) = Simplify.

Simplify a complex fraction (Method 2) EXAMPLE 6 Simplify a complex fraction (Method 2) 5 x + 4 1 + 2 x Simplify: SOLUTION The LCD of all the fractions in the numerator and denominator is x(x + 4). 5 x + 4 1 + 2 x 5 x + 4 1 + 2 x = x(x+4) Multiply numerator and denominator by the LCD. x + 2(x + 4) 5x = Simplify. 5x 3x + 8 = Simplify.

Solve a rational equation by cross multiplying EXAMPLE 1 Solve a rational equation by cross multiplying Solve: 3 x + 1 = 9 4x + 1 3 x + 1 = 9 4x + 1 Write original equation. 3(4x + 5) = 9(x + 1) Cross multiply. 12x + 15 = 9x + 9 Distributive property 3x + 15 = 9 Subtract 9x from each side. 3x = – 6 Subtract 15 from each side. x = – 2 Divide each side by 3. The solution is –2. Check this in the original equation. ANSWER