Download presentation

1
**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

2
**Standardized Test Practice**

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

3
**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.

4
**Multiply rational expressions**

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

5
**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

6
**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

7
**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

8
**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.

9
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

10
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

11
**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).

12
**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.

13
**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.

14
**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.

15
**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.

16
**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

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google