Adding and Subtracting Polynomials

Transcription

1 3.3 Adding and Subtracting Polynomials 3.3 OBJECTIVES 1. Add two polynomials 2. Subtract two polynomials NOTE The plus sign between the parentheses indicates the addition. Addition is always a matter of combining like quantities (two apples plus three apples, four books plus five books, and so on). If you keep that basic idea in mind, adding polynomials will be easy. It is just a matter of combining like terms. Suppose that you want to add 5x 2 3x 4 and 4x 2 5x 6 Parentheses are sometimes used in adding, so for the sum of these polynomials, we can write (5x 2 3x 4) (4x 2 5x 6) Now what about the parentheses? You can use the following rule. Rules and Properties: Removing Signs of Grouping Case 1 If a plus sign ( ) or nothing at all appears in front of parentheses, just remove the parentheses. No other changes are necessary. NOTE Just remove the parentheses. No other changes are necessary. Now let s return to the addition. (5x 2 3x 4) (4x 2 5x 6) 5x 2 3x 4 4x 2 5x 6 NOTE Note the use of the associative and commutative properties in reordering and regrouping. NOTE Here we use the distributive property. For example, 5x 2 4x 2 (5 4)x 2 9x 2 NOTE We call this the horizontal method because the entire problem is written on one line is the horizontal method is the vertical method. Collect like terms. (Remember: have the same variables raised to the same power). (5x 2 4x 2 ) (3x 5x) (4 6) Combine like terms for the result: 9x 2 8x 2 As should be clear, much of this work can be done mentally. You can then write the sum directly by locating like terms and combining. Example 1 illustrates this property. Example 1 Combining Like Terms Add 3x 5 and 2x 3. the sum. (3x 5) (2x 3) 3x 5 2x 3 5x 2 271

2 272 CHAPTER 3 POLYNOMIALS CHECK YOURSELF 1 Add 6x 2 2x and 4x 2 7x. The same technique is used to find the sum of two trinomials. Example 2 Adding Polynomials Using the Horizontal Method Add 4a 2 7a 5 and 3a 2 3a 4. the sum. (4a 2 7a 5) (3a 2 3a 4) REMEMBER Only the like terms are combined in the sum. 4a 2 7a 5 3a 2 3a 4 7a 2 4a 1 CHECK YOURSELF 2 Add 5y 2 3y 7 and 3y 2 5y 7. Example 3 Adding Polynomials Using the Horizontal Method Add 2x 2 7x and 4x 6. the sum. (2x 2 7x) (4x 6) 2x 2 7x 4x 6 These are the only like terms; 2x 2 and 6 cannot be combined. 2x 2 11x 6 CHECK YOURSELF 3 Add 5m 2 8 and 8m 2 3m. As we mentioned in Section 3.1 writing polynomials in descending-exponent form usually makes the work easier. Look at Example 4. Example 4 Adding Polynomials Using the Horizontal Method Add 3x 2x 2 7 and 5 4x 2 3x.

3 ADDING AND SUBTRACTING POLYNOMIALS SECTION the polynomials in descending-exponent form, then add. ( 2x 2 3x 7) (4x 2 3x 5) 2x 2 12 CHECK YOURSELF 4 Add 8 5x 2 4x and 7x 8 8x 2. Subtracting polynomials requires another rule for removing signs of grouping. Rules and Properties: Removing Signs of Grouping Case 2 If a minus sign ( ) appears in front of a set of parentheses, the parentheses can be removed by changing the sign of each term inside the parentheses. The use of this rule is illustrated in Example 5. Example 5 Removing Parentheses In each of the following, remove the parentheses. NOTE This uses the distributive property, because (2x 3y) ( 1)(2x 3y) 2x 3y (a) (2x 3y) 2x 3y Change each sign to remove the parentheses. (b) m (5n 3p) m 5n 3p Sign changes. (c) 2x ( 3y z) 2x 3y z Sign changes. CHECK YOURSELF 5 (a) (3m 5n) (b) (5w 7z) (c) 3r (2s 5t) (d) 5a ( 3b 2c) Subtracting polynomials is now a matter of using the previous rule to remove the parentheses and then combining the like terms. Consider Example 6. Example 6 NOTE The expression following from is written first in the problem. Subtracting Polynomials Using the Horizontal Method (a) Subtract 5x 3 from 8x 2. (8x 2) (5x 3) 8x 2 5x 3 Recall that subtracting 5x is the same as adding 5x. 3x 5 Sign changes.

4 274 CHAPTER 3 POLYNOMIALS (b) Subtract 4x 2 8x 3 from 8x 2 5x 3. (8x 2 5x 3) (4x 2 8x 3) 8x 2 5x 3 4x 2 8x 3 4x 2 13x 6 Sign changes. CHECK YOURSELF 6 (a) Subtract 7x 3 from 10x 7. (b) Subtract 5x 2 3x 2 from 8x 2 3x 6. Again, writing all polynomials in descending-exponent form will make locating and combining like terms much easier. Look at Example 7. Example 7 Subtracting Polynomials Using the Horizontal Method (a) Subtract 4x 2 3x 3 5x from 8x 3 7x 2x 2. (8x 3 2x 2 7x) ( 3x 3 4x 2 5x) =8x 3 2x 2 7x 3x 3 4x 2 5x Sign changes. 11x 3 2x 2 12x (b) Subtract 8x 5 from 5x 3x 2. (3x 2 5x) (8x 5) 3x 2 5x 8x 5 3x 2 13x 5 Only the like terms can be combined. CHECK YOURSELF 7 (a) Subtract 7x 3x 2 5 from 5 3x 4x 2. (b) Subtract 3a 2 from 5a 4a 2. If you think back to addition and subtraction in arithmetic, you ll remember that the work was arranged vertically. That is, the numbers being added or subtracted were placed under one another so that each column represented the same place value. This meant that in adding or subtracting columns you were always dealing with like quantities.

5 ADDING AND SUBTRACTING POLYNOMIALS SECTION It is also possible to use a vertical method for adding or subtracting polynomials. First rewrite the polynomials in descending-exponent form, then arrange them one under another, so that each column contains like terms. Then add or subtract in each column. Example 8 Adding Using the Vertical Method Add 2x 2 5x, 3x 2 2, and 6x 3. 2x 2 5x 3x 2 2 6x 3 5x 2 x 1 CHECK YOURSELF 8 Add 3x 2 5, x 2 4x, and 6x 7. The following example illustrates subtraction by the vertical method. Example 9 Subtracting Using the Vertical Method (a) Subtract 5x 3 from 8x 7. 8x 7 ( ) 5x 3 3x 4 To subtract, change each sign of 5x 3 to get 5x 3, then add. 8x 7 5x 3 3x 4 (b) Subtract 5x 2 3x 4 from 8x 2 5x 3. 8x 2 5x 3 ( ) 5x 2 3x 4 3x 2 8x 7 8x 2 5x 3 5x 2 3x 4 3x 2 8x 7 To subtract, change each sign of 5x 2 3x 4 to get 5x 2 3x 4, then add. Subtracting using the vertical method takes some practice. Take time to study the method carefully. You ll be using it in long division in Section. 3.6.

6 276 CHAPTER 3 POLYNOMIALS CHECK YOURSELF 9 Subtract, using the vertical method. (a) 4x 2 3x from 8x 2 2x (b) 8x 2 4x 3 from 9x 2 5x 7 CHECK YOURSELF ANSWERS 1. 10x 2 5x 2. 8y 2 8y 3. 13m 2 3m x 2 11x 5. (a) 3m 5n; (b) 5w 7z; (c) 3r 2s 5t; (d) 5a 3b 2c 6. (a) 3x 10; (b) 3x (a) 7x 2 10x; (b) 4a 2 2a x 2 2x (a) 4x 2 5x; (b) x 2 9x 10

7 Name 3.3 Exercises Section Date Add. 1. 6a 5 and 3a x 3 and 3x b 2 11b and 5b 2 7b 4. 2m 2 3m and 6m 2 8m 5. 3x 2 2x and 5x 2 2x 6. 3p 2 5p and 7p 2 5p 7. 2x 2 5x 3 and 3x 2 7x d 2 8d 7 and 5d 2 6d b 2 8 and 5b x 3 and 3x 2 9x 11. 8y 3 5y 2 and 5y 2 2y 12. 9x 4 2x 2 and 2x a 2 4a 3 and 3a 3 2a m 3 2m and 6m 4m x 2 2 7x and 16. 5b 3 8b 2b 2 and 5 8x 6x 2 3b 2 7b 3 5b Remove the parentheses in each of the following expressions, and simplify when possible. 17. (2a 3b) 18. (7x 4y) ANSWERS a (2b 3c) 20. 7x (4y 3z) 21. 9r (3r 5s) m (3m 2n) 23. 5p ( 3p 2q) 24. 8d ( 7c 2d)

8 ANSWERS Subtract. 25. x 4 from 2x x 2 from 3x m 2 2m from 4m 2 5m 28. 9a 2 5a from 11a 2 10a 29. 6y 2 5y from 4y 2 5y 30. 9n 2 4n from 7n 2 4n x 2 4x 3 from 3x 2 5x x 2 2x 4 from 5x 2 8x a 7 from 8a 2 9a 34. 3x 3 x 2 from 4x 3 5x 35. 4b 2 3b from 5b 2b y 3y 2 from 3y 2 2y 37. x 2 5 8x from 38. 4x 2x 2 4x 3 from 3x 2 8x 7 4x 3 x 3x 2 Perform the indicated operations. 39. Subtract 3b 2 from the sum of 4b 2 and 5b Subtract 5m 7 from the sum of 2m 8 and 9m Subtract 3x 2 2x 1 from the sum of x 2 5x 2 and 2x 2 7x Subtract 4x 2 5x 3 from the sum of x 2 3x 7 and 2x 2 2x Subtract 2x 2 3x from the sum of 4x 2 5 and 2x Subtract 5a 2 3a from the sum of 3a 3 and 5a Subtract the sum of 3y 2 3y and 5y 2 3y from 2y 2 8y. 46. Subtract the sum of 7r 3 4r 2 and 3r 3 + 4r 2 from 2r 3 +3r Add, using the vertical method w 2 + 7, 3w 5, and 4w 2 5w 48. 3x 2 4x 2, 6x 3, and 2x x 2 3x 4, 4x 2 3x 3, and 2x 2 x x 2 2x 4, x 2 2x 3, and 2x 2 4x 3 278

9 ANSWERS Subtract, using the vertical method a 2 2a from 5a 2 3a 52. 6r 3 4r 2 from 4r 3 2r x 2 6x 7 from 8x 2 5x x 2 4x 2 from 9x 2 8x x 2 3x from 8x x 2 6x from 9x 2 3 Perform the indicated operations. 57. [(9x 2 3x 5) (3x 2 2x 1)] (x 2 2x 3) [(5x 2 2x 3) ( 2x 2 x 2)] (2x 2 3x 5) 60. Find values for a, b, c, and d so that the following equations are true ax 4 5x 3 x 2 cx 2 9x 4 bx 3 x 2 2d (4ax 3 3bx 2 10) 3(x 3 4x 2 cx d) x 2 6x Geometry. A rectangle has sides of 8x 9 and 6x 7. Find the polynomial that represents its perimeter. 6x 7 8x Geometry. A triangle has sides 3x 7, 4x 9, and 5x 6. Find the polynomial that represents its perimeter. 5x 6 3x 7 4x Business. The cost of producing x units of an item is C x. The revenue for selling x units is R 90x x 2. The profit is given by the revenue minus the cost. Find the polynomial that represents profit. 64. Business. The revenue for selling y units is R 3y 2 2y 5 and the cost of producing y units is C y 2 y 3. Find the polynomial that represents profit. 279

10 ANSWERS a. b. c. d. e. f. g. h. Multiply. (a) x 5 x 7 (b) y 8 y 12 (c) 2a 3 d 4 (d) 3m 5 m 2 (e) 4r 5 3r (f) 6w 2 5w 3 (g) ( 2x 2 )(8x 7 ) (h) ( 10a)( 3a 5 ) Answers Getting Ready for Section 3.4 [Section 1.7] 1. 9a b 2 18b 5. 2x x 2 2x b 3 5b y 3 2y 13. a 3 4a x 2 x a 3b 19. 5a 2b 3c 21. 6r 5s 23. 8p 2q 25. x m 2 3m 29. 2y x 2 x a 2 12a b 2 8b 37. 2x b x x 2 5x y 2 8y 47. 6w 2 2w x 2 x 51. 2a 2 5a 53. 3x 2 x 55. 3x 2 3x x 2 3x a 3, b 5, c 0, d x x 2 65x 150 a. x 12 b. y 20 c. 2a 7 d. 3m 7 e. 12r 6 f. 30w 5 g. 16x 9 h. 30a 6 280