Elementary Algebra and Geometry for Schools

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Algebra. Some Fundamental Operations. Order of operations.

A term is an algebraic expression which is not separated into parts by a plus or a minus sign. If an algebraic expression is separated into two or more parts by plus or minus signs, each of these parts is a term of the expression. For example:

1.The following expressions contain one term:

2.The following expressions contain two terms:

3.The following expressions contain three terms:

When an algebraic expression consists of just one term, it is called a monomial; the expressions

are examples of monomials. When an algebraic expression consists of two terms, it is called a binomial; the expressions

are examples of binomials. When an algebraic expression consists of three terms, it is called a trinomial; the expressions

are examples of trinomials. An algebraic expression consisting of more than a single term is called a polynomial. Binomials and trinomials are, therefore, polynomials.

ORAL EXERCISES

State whether each of the following is a monomial, binomial, or trinomial:

Answers : 1. binomial. 2. monomial. 3. trinomial. 4. binomial. 6. monomial. 6. trinomial. 7. binomial. 8. binomial. 9. trinomial. 10. monomial. 11. binomial. 12. monomial. 13. binomial. 14. binomial. 16. trinomial. 16. trinomial.

Order of operations. When several numbers are connected by the signs +, -, X, and ÷, there is uncertainty as to the value of the expression until the correct order of operations has been determined. Thus the value of

70 ÷ 5 - 4 X 3

might be 30, 210, or 2, depending upon the order in which the operations are performed. To avoid this uncertainty, mathematicians have agreed that multiplications and divisions shall be performed before additions and subtractions; consequently, the correct value of the above expression is 2. Since we have learned that a plus or a minus sign separates an expression into terms

we can restate our order of operations.

• Simplify terms before adding them to or subtracting them from other terms.

To find the value of the expression 70 ÷ 5 - 4 X 3, we recognize that this is a binomial and that the minus sign separates the expression into two terms. We now perform the indicated division in the first term and the indicated multiplication in the second term, and then we subtract the second term from the first. Thus: 70 ÷ 5 is 14; 4 X 3 is 12; and finally, 14 -12 is 2.

The binomial 2a +b indicates that the term 2a is to be added to the term b. If we are given that a = 5 and b = 4, then the numerical value of 2a + b is 2·5 + 4; that is, 10 + 4 or 14. Notice that we simplified the term 2 . 5 before adding it to the term 4.

ILLUSTRATIVE EXAMPLES: Order of operations

EXERCISES: Order of operations; evaluating polynomials

Find the numerical value of each of the following:

Answers : 1. 27. 2. 3. 3. 8. 4. 6. 5. 11. 6.1.

If a = 4, b = 9, x = 6, and y = 2, find the numerical value of each of the following algebraic expressions:

Answers : 7. 17. 8. 5. 9. 9. 10. 25. 11. 12. 12. 29. 13. 6. 14. 4. 15. 10. 16. 34. 17. 3. 18. 5. 19. 31. 20. 9. 21. 20. 22. O. 23. 6.

 Factors. When two or more numbers are multiplied together to form a product, each of the numbers is a factor of that product. Thus since the product of 3 and 7 is 21, each of the numbers 3 and 7 is a factor of 21. In the algebraic expression 4ab, each of the numbers 4, a, and b is a factor of the expression.

To factor a number means to find two or more numbers which when multiplied together will equal the original number. Thus the possible pairs of factors of the number 30 are: 1 and 30; 2 and 15; 3 and 10; 5 and 6. Similarly, to factor an algebraic expression is to find two or more quantities whose product gives the original expression. Possible pairs of factors of the expression 6ab are: 6a and b; 2a and 3b; a and 6b; 6 and ab; 3a and 2b.

 Coefficient. If an expression is in factored form, each factor is said to be the coefficient of the remaining factors. Thus in the expression 3ab, the coefficient of ab is 3, the coefficient of b is 3a, and the coefficient of a is 3 b. If the expression contains a single numerical factor, as in the expression 3ab, the number, here 3, is said to be the numerical coefficient. You must remember that if no number appears in the expression, there is still a numerical coefficient of 1, but the 1 need not be written. Thus the expression xy has a numerical coefficient of 1, but we need not indicate this by writing 1xy or by calling it "one xy."

Order of factors in a term. In arithmetic, if we are asked to multiply three or more numbers together, we know that the order in which we perform the multiplications does not matter. Thus the product of the numbers 4, 5, and 6 is 120 whether we multiply the numbers in the order 4 X 5 X 6 or 4 X 6 X 5 or 5 X 6 X 4. In algebra, as in arithmetic, the order in which we multiply three or more numbers (including literal numbers) does not matter. Thus if x represents some number, and y represents some other number, the product of the three numbers 5, x, and y is the same, regardless of the order in which the multiplication is performed. That is, the terms 5xy, or 5yx, or xy5 all have the same meaning. It is customary, however, to write first the numerical coefficient of a term; then the numerical coefficient is followed by the letters in alphabetical order. Thus we would write 5xy and not 5yx; we would write 6ax and not a6x; we would write xyz and not yxz.

EXERCISES: Order of factors in a term.

Arrange the factors of each of the following terms in their customary order:

Answers : 1. 7 ab. 2. 2bc. 3. 5ay. 4. 6ab. 5. 8xyz. 6. abc. 7. 4mx. 8. 3rst. 9. 7axy. 10. 9abxy. 11. 2abr. 12. 3bxy.

EXERCISES: Completion test

On another sheet of paper, write the numbers of the incomplete statements in a column. Opposite the number of each incomplete statement, write the number or letter(s) for each blank which will complete the statement and make it true.

Answers : 1. 7. 2. 6. 3. c. 4. 11x . 5. 4. 6. 9yz. 7. 13b. 8. 3ab. 9. 6xy. 10. 1. 11. 4, 3. 12. 15x. 13. 1/2. 14. 5xy. 15. 6ab.

Multiplication and division of monomials. In arithmetic, to multiply the expression 3 X 4 by 2, we simply write 2 X 3 X 4. This multiplication, which can be done in any order, gives the product 24. In multiplying the expression 3 X 4 by 2 we do not multiply both the 3 and the 4 by 2; to do so would give 6 X 8 or 48 which is incorrect.

 Similarly, in algebra, to multiply 5a by 3, we write 3.5a. To simplify this expression we multiply 3 by 5, and then multiply this product, 15, by a, obtaining 15a To divide the expression 6 X 4 by 2, we divide either the 6 by 2 or we divide the 4 by 2, but we do not divide both the 6 and the 4 by 2. Since the expression 6 X 4 has the value 24, and since 24 divided by 2 equals :12, we observe that both of the adjacent procedures give the correct answer. To divide 15m by 5, we divide 15 by 5 and then multiply the quotient, 3, by m.

ILLUSTRATIVE EXAMPLES: Multiplication and division of monomials

EXERCISES: Multiplication and division of monomials

Answers : 1. 6x. 2. 21 a. 3. 14xy. 4. 6ab. 5. 9ab. 6. 12xy. 7. 20ab 8. 24mn. 9. 14. 10. 2z. 11. 4y. 12. 7. 13. 13. 14. 17. 15. 6x. 16. 15xy. 17. 35ab. 18. 36ax. 19. 48xy. 20. 72ab. 21. 7x. 22. 8. 23. 8x. 24. 11 b. 25. 8y. 26. 8y. 27. 45abxy. 28. 140ab. 29. 10x. 30. 76y. 31. 8a. 32. 48. 33. 55/4 a. 34. 152/9 xy. 35. 28x. 36. 54.

Use of parentheses. Parentheses ( ) are used for two purposes:

1. To indicate multiplication. Thus, (3) (5) and 3(5) both mean that 3 is to be multiplied by 5.

2. To indicate that the expression within them is to be considered as a single quantity. Thus, 2(7 + 5) means that the binomial 7 + 5 is to be treated as a single quantity, 12, and multiplied by 2.

EXERCISES: Evaluating algebraic expressions.

Evaluate each of the following:

Answers : 1. 12. 2. 4. 3. 1. 4. 18. 5. 13. 6. 7. 7. 10. 8. 6. 9. 7. 10. 23. 11. 7. 12. 49. 13. 2(a + b). 14. x + 3(c + d). 15. y -2(r + s).

Removing parentheses. As we have seen, the expression 2(5 +3) means twice the sum of 5 and 3, or 16. Here we first added the two numbers inside the parentheses and then multiplied this sum by 2. We can obtain the same correct answer if we first multiply both the 5 and the  3 by 2, and then add the results. That is

This gives us a method of removing parentheses when some of the numbers are represented by letters whose sum can only be indicated. We may illustrate this method of removing parentheses as follows:

ILLUSTRATIVE EXAMPLES: Removing parentheses

1. Remove the parentheses in the expression 2(a +b) by performing the indicated multiplication.

Solution.

2(a +b)

2. Remove the parentheses in the expression 3x(7 -2y) by performing the indicated multiplication.

Solution.

3x(7 -2y)

EXERCISES: Removing parentheses.

Remove the parenthes in each of the following exercises by performing the indicated multiplication:

Answers : 1. 2x + 2y. 2. 5a -5b. 3. 3m + 3n. 4. 6p -6q. 5. 4c + 4d. 6. 10x + 5y. 7. 3a -6b. 8. 6x + 2y. 9. 14c -7d. 10. 24p + 8q. 11. 8x + 20y. 12. 27a -8b. 13. 30c -24d. 14. ax + ay. 15. 2ax -2ay. 16. 2ax -ay. 17. 2ax + 2bx. 18. 3ay + 3by. 19. 2mx -2nx. 20. 3ax + ay. 21. 6ax -15ay.

Exponents. In the first chapter we reviewed the formula A = lw for finding the area of a rectangle. Applying this formula to find the area of a square whose length and width are both 5, we have: A = 5. 5, or A = 25.

A convenient way of indicating 5 . 5 is to write 52 (read "five to the second power" or "five squared "), where the small number 2, called an exponent, means that the number 5, called the base, is to be taken 2 times as a factor. Similarly, we can indicate 5 . 5 . 5 as 53 (read "five to the third power" or "five cubed"), where the exponent 3 means 5 that the number 5 is to be taken 3 times as a factor. Similarly, x2 means x.x; y3 means y . y . y; a4 means a . a . a . a.

Since we are considering here only exponents that are integers (whole numbers) such as 1, 2, 3, 4, etc., we defme an integral exponent as follows:

• An exponent is a small number written to the right of, and above, a second number called the base. The exponent indicates how many times the base is used as a factor.

If an exponent of a number or a letter is 1, the exponent is not generally indicated. By the above definition, 51 (read" five to the first power") means that the number 5 is to be taken once as a factor; that is, 51 is 5. Similarly, x1 is x; y1 is y and a1 is a.

ILLUSTRATIVE EXAMPLES: The use of exponents

1. Write the expression 3·3·3·3·3 as 3 raised to a power.

Solution. We wish to indrcate that the number 3 is to be taken five times 3·3·3·3·3 as a factor; that is, the exponent is 35 to be 5.

2. Write the expression 3a2b3c without the use of exponents.

Solution. The exponents 2 and 3, respectively, tell us that a is to be taken twice as a factor and b is to be taken three times as a factor; the exponent of the number 3 and of the letter c are each understood to be 1.

3a2b3c = 3aabbbc

EXERCISES: Exponents

In 1-12. Write each expression by using exponents.

In 13-24; write each expression without using exponents.

Answers : 1. x3 . 2. y2. 3. a4 .4. a2b3 .5. x2y3. 6. a2b3c. 7. x5. 8. 85. 9. 3a2b3c4 . 10. 2x3y2. 11. 5x2y3z2. 12. 6a3b2c5 13. aabbb. 14. xxyy. 15. mmmnn. 16. 2abbc. 17. 7xxxxx. 18. 5xxyzz. 19. 6aaabb. 20. xxxyyz. 21. 8ppppqq. 22. mmnnnpp. 23. 9xyyyyy. 24. aaaabbbcc.

Powers of numbers. You have learned that "3 raised to the fourth power" is written 34 and means 3 . 3 . 3 . 3. Performing this indicated

multiplication, we find the value of 34 is 81. Similarly, we can find the value of any number raised to a power.

EXERCISES: Powers of numbers

Find the numerical value of each of the following:

Answers : 1. 9. 2. 8. 3. 25. 4. 49. 5. 81. 6. 36. 7. 512. 8. 121. 9. 64. 10. 18. 11. 400. 12. 343. 13. 32. 14. 1. 15. 16. 16. 225. 17. 10,000. 18. 27. 19. 216. 20. 64. 21. 1/4. 22. 1/8 . 23. 1/16 . 24. 4/9 . 25. 27/8

A table of powers of numbers. In your earlier work in arithmetic you learned multiplication tables which greatly helped you in your work with numbers. We can now prepare a "power table" so that you will be able to do further work with numbers more quickly and more easily.

The table below shows the numbers from 1 to 5 raised to the first five powers. You should study this table until you have learned the values of the numbers 1 to 5 raised to the first, second, third, fourth, and fifth powers.

ORAL EXERCISES: Powers of numbers

Without looking at the above table, practice your knowledge of the table by giving the answer to each of the following exercises:

Answers : 1. 9. 2. 125. 3. 32. 4. 16 . 5. 1. 6. 16. 7. 3. 8. 25. 9. 1. 10. 1024. 11. 5. 12. 1. 13. 64. 14. 8. 15. 243. 16. 2. 17. 1. 18. 625. 19. 4. 20. 81. 21. 27. 22. 256. 23. 3125. 24. 4. 25. 1.

ILLUSTRATIVE EXAMPLES: Evaluating expressions containing exponents

EXERCISES: Evaluating algebraic expressions

Answers : 1. 9. 2. 64. 3. 32. 4. 18. 5. 36. 6. 50. 7. 100. 8. (a) 27; (b) 81; (c) 36; (d) 135; (e) 3; (f) 1/3; (g) 15; (h) 36. 9. (a) 32; (b) 125; (c) 75; (d) 250; (e) 25; (f) 40; (g) 25; (h) 100. 10. 13. 11. 24 . 12. 92. 13. 96. 14. 98. 15. 22. 16. 7. 17. 90. 18. 6. 19. 12. 20. 75. 21. 37.