Section 6.4 Adding & Subtracting Like Fractions

Transcription

1 Section 6.4 Adding & Subtracting Like Fractions ADDING ALGEBRAIC FRACTIONS As you now know, a rational expression is an algebraic fraction in which the numerator and denominator are both polynomials. Just as in arithmetic, algebraic fractions with the same denominator are called like fractions. As you know from arithmetic, to add two fractions they must have the same denominator, they must be like fractions. To add fractions (of any type), they must have a common denominator: a c + b c a + b c Example 1: Add; simplify your result, if possible: Arithmetic Example Algebraic Example x 2 4 The fractions have the same denominator. 2x x 2x Combine the fractions by adding the numerators. x 2 + 3x 2x Factor the numerator and denominator. x (x + 3) 2 (x + 3) 7 2 Reduce by canceling common factors. x 2 Exercise 1 Add; simplify your result, if possible: b) 4x 2x x + 3 c) x 2 x x x 2 4 Adding and Subtracting Like Fractions page 6.4-1

2 The steps for adding rational expressions are as follows: To Add Rational Expressions with Common Denominators 1. add the numerators together and put the sum over the common denominator; 2. if necessary, combine like terms and write the numerator in descending order; 3. if possible, factor both the numerator and denominator; and 4. if possible, simplify the fraction by canceling common factors. Let s apply these steps to some more examples Example 2: Add these fractions with common denominators; simplify your result, if possible: x 10x x b) x 2 4x + 5x 4x Procedure: Combine these into one fraction; the single fraction will have the same (common) denominator as the two original fractions. The fractions have common denominators. x 10x x b) x 2 4x + 5x 4x Combine them into one fraction. x x x 2 + 5x 4x Factor both numerator and denominator. 1(x + 8) 10x x (x + 5) 4 x Simplify, if possible. (x + 8) 10x (x + 5) 4 This fraction can't be simplified. This fraction cannot be simplified any further. Adding and Subtracting Like Fractions page 6.4-2

3 Example 3: Add these fractions with common denominators; simplify your result, if possible: x 2 10 x 2 2x x 2 2x 15 b) x 2 + 3x 1 x 2 6x x 8 x 2 6x + 5 Procedure: Combine these into one fraction; the single fraction will have the same (common) denominator as the two original fractions. The fractions have common denominators. x 2 10 x 2 2x x 2 2x 15 b) x 2 + 3x 1 x 2 6x x 8 x 2 6x + 5 Combine them into one fraction. x x 2 2x 15 Combine like terms. x 2 + 3x 1 + 5x 8 x 2 6x + 5 x 2 9 x 2 2x 15 Factor the numerator and denominator. x 2 + 8x 9 x 2 6x + 5 (x 3)(x + 3) (x 5)(x + 3) (x 1)(x + 9) (x 5)(x 1) Cancel any common factors. (x 3) (x 5) (x + 9) (x 5) Exercise 2 Add these fractions with common denominators; simplify your result, if possible: 5c 4c + 1 4c b) 3w 2w w 1 c) 4x + 3 x 2 + 5x x + 7 x 2 + 5x + 6 y 2 6 y 2 3y y 2 3y 10 Exercise 3 Add these fractions with common denominators; simplify your result, if possible: Adding and Subtracting Like Fractions page 6.4-3

4 5a 1 a a + 7 a + 1 b) 3y y y 2 4 c) 3c + 2 c 2 c + c 6 c 2 c x 2 x x x 2 9 e) x 2 + 3x x 2 + x 6 + x 12 x 2 + x 6 f) m 2 m 2 7m m 12 m 2 7m + 12 SUBTRACTING ALGEBRAIC FRACTIONS Adding and Subtracting Like Fractions page 6.4-4

5 The rules for subtracting fractions are virtually the same as for adding fractions: To subtract, fractions (of any type) must have a common denominator: a c b c a b c Example 4: Subtract; simplify your result, if possible: x 3x 6 3x b) x x x 2 16 Procedure: Follow the guidelines presented below: The fractions have common denominators. x 3x 6 3x b) x x x 2 16 Combine them into one fraction. x 6 3x x 4 x 2 16 Factor both numerator and denominator. 1(x 6) 1 (x 4) 3x This fraction simplifies ()(x 4) (x 6) 3x This fraction doesn t simplify. 1 () Subtraction should not, however, be taken lightly. Subtraction has always been more challenging than addition, and we need to be especially careful with it when working with polynomials and algebraic fractions. When subtracting polynomials, for example, we need to make sure that the subtraction sign applies to the entire second polynomial; we do so by multiplying it by - 1. Let s practice subtracting polynomials. Example 5: Subtract these polynomials. (3) (x 6) b) (x 2 3x) (8) Procedure: Be sure to distribute the minus sign through to the second polynomial (it s the same as multiplying through by -1). Combine like terms. 1 (3) 1 (x 6) b) 1 (x 2 3x) 1 (8) 3 x + 6 x 2 3x 8x 4 2x + 10 x 2 11x 4 When subtracting algebraic fractions, we need to make sure that we are subtracting the entire fraction; that means we need to make sure that the entire numerator of the second fraction is subtracted from the first. We use parentheses to keep things in order. Adding and Subtracting Like Fractions page 6.4-5

6 Example 6: Subtract; simplify your result, if possible: x + 6 x + 2 x + 1 x + 2 b) 2x 5 x 6 c) 2x 2 + 2x 5 x 2 4x + 3 x 2 x 1 x 2 4x + 3 3x 2 6x 2 + 7x x x 2 + 7x + 2 Procedure: Use parentheses around each numerator and subtract as if subtracting polynomials. x + 6 x + 2 x + 1 x + 2 b) 2x 5 x 6 1 (x + 6) 1 (x + 1) x + 2 x + 6 x 1 x (2x 5) 1 (x 6) 2x 5 x x + 2 x + 1 c) 2x 2 + 2x 5 x 2 4x + 3 x 2 x 1 x 2 4x + 3 3x 2 6x 2 + 7x x x 2 + 7x (2x 2 + 2x 5) 1 (x 2 x 1) x 2 4x + 3 3x 2 1 (13x + 10) 6x 2 + 7x + 2 2x 2 + 2x 5 x 2 + x + 1 x 2 4x + 3 3x 2 13x 10 6x 2 + 7x + 2 x 2 + 3x 4 x 2 4x + 3 Use the Factor Game to factor these. ()(x 1) (x 3)(x 1) (3x + 2)(x 5) (3x + 2)(2x + 1) x 3 x 5 2x + 1 Exercise 4 Subtract; simplify your result, if possible: Adding and Subtracting Like Fractions page 6.4-6

7 3y + 4 2y 1 5 2y 1 b) 3x 3 x c) 3w 4w 8 6 4w 8 5a 1 a 2 2a + 5 a 2 e) 3c + 2 c 2 + 4c c 6 c 2 + 4c f) x 2 x 2 9 3x x 2 9 g) m 2 m 2 7m m 20 m 2 7m + 12 h) y 2 3y y 2 3y 10 y + 12 y 2 3y 10 NEGATIVES IN THE DENOMINATOR Adding and Subtracting Like Fractions page 6.4-7

8 Sometimes a fraction has a negative in the denominator, such as 3x - 8 or 4x - (x + 2). It is common practice to rewrite such fractions so that the negative is moved to the numerator. The reason we re allowed to move a 6 negative from the denominator to the numerator is illustrated here. Consider the fraction - 3, which is the same as 6 (- 3). Each of these has a value of - 2. So, since numerator and and even Similarly: we can, in effect, move the negative from the denominator into the 3 - x could be rewritten as - 3 x Also: 5 - (x 3) - 5 (x 3) Example 7: Rewrite each of these so that the negative is no longer in the denominator. 8-7 b) 4x - (x 5) c) (x + 2) - (x 5) x (x 5) - (x + 2) (x 5) - x 2 (x 5) Exercise 5 Rewrite each of these so that the negative is no longer in the denominator. Simplify. 5-2x b) 3y - (y + 4) c) - 2x - (x 1) 3x 2 - (x 8) e) m (m + 4) f) - 4w 5 - (3w 1) When adding or subtracting, if a fraction s denominator has a negative, that negative should be moved to the numerator. Adding and Subtracting Like Fractions page 6.4-8

9 Example 8: Add or subtract. Move any negative from the denominator into the numerator, then continue. (After the move the fractions will become like fractions.) b) x x (x 5) c) x + 3 x 2 + x 1 - (x 2) x (x 5) -4 (x 5) x + 3 (x 2) + - (x 1) (x 2) 6 + (-8) 7 x (-4) x 5 x + 3 (x 2) + - x + 1 (x 2) x 5 x (- x) + 1 (x 2) 4 (x 2) Exercise 6 Add or subtract. First, move any negative from the denominator into the numerator. Simplify if possible. 3 2x + 5-2x b) 5 3y - 4-3y c) 2y y (y + 4) 3x x (x 2) e) x + 1 x 8 + 3x 2 - (x 8) Answers to each Exercise Adding and Subtracting Like Fractions page 6.4-9

10 Section 6.4 Exercise 1: 3 2 b) 2 c) x (x 2) Exercise 2: (5c + 1) 4c b) (3w + 5) (2w 1) c) 5 (x + 3) (y 2) (y 5) Exercise 3: 6 b) 3 (y 2) c) 4 c x (x 3) e) (x + 6) (x + 3) f) (m + 4) (m 4) Exercise 4: (3y 1) (2y 1) b) (2x 9) (3) c) e) 2 c f) x (x + 3) g) (m 5) (m 3) h) (y 6) (y 5) Exercise 5: - 5 2x b) - 3y (y + 4) c) 2x (x 1) - 3x + 2 (x 8) e) - m 1 (m + 4) f) 4w + 5 (3w 1) Exercise 6: - 1 x b) 1 3y c) 2y + 3 y e) - 2x + 3 x 8 Section 6.4 Focus Exercises Adding and Subtracting Like Fractions page

11 1. Add; simplify your result, if possible: 2y 7y + 5 7y b) 2x + 5 3x 6 3x c) 3p p p + 5 3w + 1 2w w + 7 2w + 4 e) 2x 6 x 2 4x + x 6 x 2 4x f) 4m + 3 m m m 2 25 g) 2x + 5 x 2 + x 6 + x 2 + x 6 h) y 2 3y y 2 3y 28 + y y y 2 3y 28 i) 2v 2 + 7v 6 v v2 + 5v + 6 v 2 16 j) 2x x 2 2x 8 + 8x x 2 x 2 2x 8 2. Subtract; simplify your result, if possible: Adding and Subtracting Like Fractions page

12 y + 6 y 5 1 y 5 b) 2x x c) w 2 w 3 9 w 3 3a + 5 a 2 4 a + 1 a 2 4 e) 4c + 1 c 2 3c 3c + 4 c 2 3c f) x x x + 6 x 2 25 g) x 2 + x x 2 11x x + 36 x 2 11x + 18 h) y 2 + 3y y 2 + 2y 24 4y + 12 y 2 + 2y 24 Adding and Subtracting Like Fractions page

13 3. Rewrite each of these so that the negative is no longer in the denominator. Simplify. 8-9y b) 4p - (3p + 1) c) - 5x - (2x 7) x 4 - (x 1) e) 3 w - (w + 6) f) - 3x 1 - (2x + 1) 4. Add or subtract, as indicated. First, move any negative from the denominator into the numerator. Simplify if possible. 8 - x b) 1-3m + 4 3m c) 2x + 1 2x + 5-2x 4x x (x + 3) e) x (x 5) + 4x 3 x 5 f) - 5y - 2y 7 2y g) x - 5x - 2 5x h) c (c 3) 2 c 3 i) y 1 y + 4 y (y + 4) Adding and Subtracting Like Fractions page