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Elementary Algebra and Geometry for Schools

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Multiplication of monomials containing exponents. We have learned that a2 means a· a and a3 means a· a· a. Thus the product (a2)(a3) means (a· a)(a· a· a) or a· a· a· a· a which we may write as a5 We notice that the product a2·a3 can be simplified to as by adding the exponents 2 and 3 to obtain the exponent 5.

Similarly x3. x4 can be simplified to x7 by adding the exponents 3 and 4 to obtain the exponent 7.

The expression b2c3 cannot be further simplified since b . b . c . c . c does not give us five b's or five c's. Thus:

  • To find the product of expressions having the same base, write that base and use as its exponent the result obtained by adding the original exponents.

ILLUSTRATIVE EXAMPLES: Multiplication of monomials containing exponents

EXERCISES: Multiplication of monomials.

Find the product of each of the following:

Answers: 1. x6. 2. a5. 3. y8. 4.b10. 5. x3 .6. p4 . 7. m9. 8. y6. 9. c11.

Answers:

 

Division of monomials that contain exponents. To simplify the 34/51 fraction we factor the numerator, 34, into 2 · 17 and we factor the denominator, 51, into 3 · 17. The fraction can now be written as

We can now divide the numerator and denominator by 17, thus obtaining the equivalent fraction 2/3.

Similarly we can simplify the fraction by factoring the a numerator into a · a and writing the fraction as a · a/a. We now divide the numerator and denominator by a obtaining the result a.

Similarly the fraction

 

We notice that can be simplified to x2 by subtracting 3 from 5 to obtain the exponent 2.

  • To find the quotient of two expressions having the same base, write that base and use as its exponent the result obtained by subtracting the exponent of the divisor from the exponent of the dividend.

 

ILLUSTRATIVE EXAMPLES: Division of monomials containing exponents

EXERCISES : Division of monomials

Find the quotient of each of the following:

Answers : 1. x4. 2. a3. 3. y. 4. b2. 5. c5 . 6. x3 . 7. y3. 8. m4. 9. 1. 10. x. 11. 4x2. 12. 5a2. 13. 5y4. 14. x. 15. 2c2 .16. 2. 17. 10a6. 18. 7m. 19. 4y2 . 20. 1.

Answers : 21. 3ab3. 22. 3xy. 23. 3x. 24. 3a. 25. 8a2y5. 26. 5m3n. 27. 0.9y2z. 28. 5r4. 29. 20. 30. 21 y2.

Like terms. We say that 5a and 7a are like terms because their literal factors are identical. Likewise, 8b and 6b are like terms because their literal factors are exactly the same. Similarly 12cd2 and 15cd2 are like terms. The terms 2a and 7b are not like terms because their literal parts are not identical. Similarly, 2ab2 and 3a2b are not like terms; 4x2 and 9x are not like terms.

  • Like terms, or similar terms, are terms that have identical literal factors.

 

Addition and subtraction of like terms. In our study of arithmetic, we learned that 3 X 7 means the sum of three 7's; 2X7means the sum of two7's. The expression 3 X 7 + 2 X 7means the sum of three7's added to the sum of two 7's; that is, five 7's in all. Similarly, in algebra, 5a means the sum of five a's, or a+a+a+a+a; 7x means the sum of seven x's; 12y means the sum of twelve y's.

Just as we can add three 7's and two 7's to get five 7's, we can add four a's and six a's to get ten a's. Similarly, 5x + 8x can be added to get 13x; the like terms 9b and 12b can be added to get 21b.

Like terms can also be subtracted. Thus fourteen y's can be subtracted from eighteen y's to get four y's.

The addition or subtraction of like terms is often called collecting like terms. Unlike terms cannot be collected (added or subtracted). Thus the sum of 2a and 3b can be expressed as 2a + 3b, but this expression cannot be further simplified since the terms are unlike.

  • To add or subtract like terms, add or subtract the numerical coefficients of the like terms and multiply this number by the literal part of the terms.

ILLUSTRATIVE EXAMPLES: Adding and subtracting like terms.

EXERCISES: Adding and subtracting like terms

In ex. 1-23 find the sum of the terms.

Answers : 1. 9x. 2. 12y. 3. 13a. 4. 8x2. 5. 10b2. 6. 30c. 7. 91y. 8. 50x. 9. 15a. 10. 21x. 11. 23y. 12. 29ab. 13. 85y2. 14. 76xy. 15. 46.7x2. 16. 106.4y. 17. 7.81a. 18. 15.00x. 19. 97.49c. 20. 7a. 21. 4/3 x. 22. 13/6 a. 23. 103.5b.

In ex. 24-43 subtract the lower term from the upper term.

Answers : 24. 3a. 25. 6x. 26. 3y2. 27. 6ab. 28. 13y. 29. 11xy. 30. 18c. 31. 0. 32. 5ab. 33. 56ab. 34. 41y3. 35. 27cd. 36. 19.87a2 . 37. 30.39ax. 38. 38.67y3. 39. 15.3p. 40. 3¼x2. 41. 11/12 y. 42. 6 7/8 mm 43. 15 2/15 a4.

ORAL EXERCISES: Recognition of like terms

Answers : 1. no. 2. yes. 3. yes. 4. yes. 5. no.

In each of ex. 6-21 select the like terms.

Answers : 6. 3a, 5a. 7. 2x, 8x. 8. 4a, 9a. 9. 14m, 8m. 10. 3xy, 5xy. 11. 6a2, a2. 12. 8a2b, 3a2b. 13. 4x2, 2x2. 14. 2b, 5b. 15. 8y2, 11y2. 16. x3, 5x3 . 17. m2, 9m2. 18. 2ab, 3ba. 19. x2y, 6yx2. 20. mn2, n2m. 21. abc, cba.

ILLUSTRATIVE EXAMPLES: Collecting like terms

1. Simplify the expression 3a + 2b +5a +7a +4b.

Solution. Given example 3a + 2b + 5a + 7a + 4b

 

EXERCISES: Collecting like terms

Simplify each of the following by collecting like terms:

Answers : 1. 5a. 2. 4x. 3. 2x. 4. 4y. 5. b. 6. 8b. 7. 10x. 8. 42y. 9. 5x. 10. 3y. 11. 11x. 12. 4m. 13. 0. 14. 12c. 15. 8d. 16. 14x + 7y. 17. 15a + 11b. 18. 3a + 10b. 19. 10y. 20. 3a + b. 21. 6x2 + 4x. 22. a2 + 15b2. 23. 8ax + 3bx. 24. 8a + 5b. 25. 17ab. 26. 7x2 + 2y2. 27. 8ab. 28. 2x + 3y + 4xy. 29. 3a3 + 8a2. 30. 2ay + 15by. 31. 3 1/4 . 32. 8 1/2 x. 33. 1 3/4 x. 34. 2 1/2 x. 35. 13/6 x . 36.  5/6 x . 37. 2 2/3 x . 38. 6.4x. 39. 17.3y. 40. 7.44 y2. 41. 11.3a.

Representing one number in terms of another. In our everyday living we frequently make comparisons between two numbers.

Bob says, "I am two years older than Mary." Bill says, "My father is three times as old as I am." Joe complains, "Don receives twenty-five cents more spending money each week than I do." In algebra we deal with similar comparisons. We find that it is often helpful to represent two quantities in terms of a single letter. Let us consider several illustrations

ILLUSTRATIVE EXAMPLES: Representing one number in terms of another

1. Bill says, "My father is three times as old as I am." If we let a represent the number of years in Bill's age, how shall we represent the number of years in his father's age?

Solution. We let the number of years in Bill's age be represented by a. Then the number of years in his father's age will be represented by 3a.

2. If x represents a certain number, how shall we represent another number that is twenty greater than x?

Solution. The other number will be represented by x + 20

ORAL EXERCISES

If x represents a number, how would you represent another number that is:

Answers: 1. 2x. 2. x + 5. 3. x - 7. 4. x/2 . 5. x + a. 6. x - b. 7. 7x. 8. x/5 . 9. ax. 10. 2ax.

EXERCISES: Multiple-choice test

Each exercise below is followed by four possible answers, only one of which is correct. Write the numbers 1-12 on another sheet of paper, and after each number, write the letter of the correct answer.

Answers : 1. c. 2. b. 3. b. 4. b. 6. d.

Answers : 6. a. 7. b. 8. c. 9. a. 10. d. 11. b. 12. c.

CHAPTER REVIEW EXERCISES

Answers : 1. 2x2 . 2. 3. 3. 1. 4. 16. 5. 2. 6. 3ac. 7. 1. 8. 3xy, 11.2 p, 17/12 ab. 9. 2xy. 10. 42. 11. 6a2bx. 12. 3a -5. 13. 2. 14. 6x. 15. 7. 16. 4x2 .17. 6x7

Answers : 18. x + 2y. 19. 3. 20. 5c.

CUMULATIVE REVIEW EXERCISES

Answers : 1. 16. 2. y =11. 3. 12 + x. 4. 18. 5. a + 2b. 6. 2a + 3b . 7. 14. 8. y3. 9. 35a2x3 .10. 3b2. 11. c, d. 12. b, d. 13. 3. 14. (a) monomial; (b) trinomial; (c) monomial; (d) binomial. 15. x + 10. 16. y = 16. 17. (a) 28; (b) 12; (c) 80. 18. (a) 6a + 3b; (b) 12x2 + 24x; (c) 2x2y2 + 2y4. 19. 2. 20. 3 + 2x.

MATCHING EXERCISES

On another sheet of paper write the numbers 1 through 12 in a vertical column. Read the first description in Column A. Find, under Column B, the item which is described. Write this word opposite the number 1 on your answer sheet. Complete the exercises in this manner.

COLUMN A COLUMN B

1. An algebraic expression of two terms.

Polynomial

2. In the algebraic expression 6ab, the number 6.

Trinomial

3. In the algebraic expression 6x3, the number 3.

Binomial

4. The result obtained when two quantities are added.

Exponent

5. Terms which have identical literal parts.

Base

6. An algebraic expression of one term.

Sum

7. Symbols used to enclose expressions which are to be considered as a single quantity.

Quotient

8. An algebraic expression of more than three terms.

Product

9. The result obtained when two quantities are multiplied.

Numerical coefficient

10. An algebraic expression of three terms.

Parentheses

11. The result obtained when one quantity is divided by another.

Like terms

12. The number 3 in the expression 32 Monomial

Answers : 1. binomial. 2. numerical coefficient. 3. exponent. 4. sum. 5. like terms. 6. monomial. 7. parentheses. 8. polynomial. 9. product. 10. trinomial. 11. quotient. 12. base.