Ex. 11. Answer the following questions.
1. What is the relationship between arithmetic and algebra? 2. In what arithmetic operations do we use numbers? 3. What do we use in algebra to represent numbers? 4. What examples of the close relationship between arithmetic and algebra can you give? 5. What is algebra? 6. What is the outstanding characteristic of algebra? 7. Name algebraic expressions you know. 8. When are two terms called similar? 9. What signs are used in algebra and what do they indicate? 10. How is the sign ( ) read? 11. What is the meaning of the multiplication sign, the equality sign and the division sign? 12. What does the expression (a + b) mean?
Ex. 12. Find the English equivalents for the following Russian words and word combinations.
1. утверждение; 2. иметь дело; 3. рассматривать; на языке; 4. вычислять; 5. подобно, таким же образом; 6. алгебраические выражения; 7. полином; 8. для удобства; 9. член; 10. трехчлен; 11. возведение в степень; 12. тогда утверждение справедливо; 13. представлять
a. then the statement is true; b. trinomial; c. raising to a power; d. deal with; e. for convenience; f. term; g. compute; h. likewise; i. in terms of; j. algebraic expressions; k. statement; l. represent; m. polynomial
Ex. 13. Give the proper English equivalents for the Russian expressions.
1. Algebra is обобщение of arithmetic. 2. In order to state the general rule we write symbols, обычные буквы, instead of particular numbers. 3. These signs have been introduced to denote отношения between numbers. 4. Algebra is the system of rules относительно the operations with numbers. 5. Particular numbers may замещать the symbols a and b. 6. All the laws of arithmetic верны for operations with letters. 7. We write symbols вместо particular numbers. 8. The square of the sum of any two numbers c and d can be вычислен by the rule (c + d)2 = c2 + 2c•d2. 9. Algebraic expressions may be given a более простая form by combining similar terms. 10. Algebraic expressions consisting of more than one term are called полиномами.
Ex. 14. Mark the following as True or False.
1. Algebra is a generalization of geometry. 2. In order to state the general rule, we write numbers instead of particular letters. 3. Algebra is the system of rules concerning the operations with numbers. 4. Since the letters used represent numbers, all the laws of arithmetic fail to hold in operations with letters. 5. The operations of addition, subtraction, multiplication, division, raising to a power and extracting roots are called algebraic expressions. 6. An expression of two terms is a trinomial. 7. As in arithmetic, the equality sign means “not equal to”.8. In finding the product of multinomials we make use of commutative law. 9. These rules cannot be easily stated as formulas in terms of letters, like the rule given above for squaring the product of two numbers. 10. The outstanding characteristic of algebra is the use of numbers to represent letters.
Ex. 15. Ask special questions.
1. A polynomial is an algebraic expression composed of one or more terms (what, how many) 2. Algebraic expressions are divided into two groups. (how many) 3. An expression 6x6 + 4x3 + 8 is of the fifth degree in x. (what) 4. If a polynomial contains but one term, it is called a monomial. (when) 5. The fundamental operations with polynomials are addition, subtraction, multiplication and division. (what) 6. If the remainder is zero, the division is exact. (when) 7. The so-called “double sign” ( ) is sometimes used. (what) 8. The equality sign (=) means “equals” or “is equal to”. (what) 9. In the operation + 10 - 10 = 0, the minus sign means that 10 is subtracted from 10 to give a zero remainder. (what) 10. We use the signs plus (+) and minus (-) to indicate addition and subtraction. (why, what for) 11. There are three requirements for an equation. (how many)
Ex. 16. Translate into Russian.
Monomials and Polynomials
1. Algebraic expressions are divided into two groups according to the last operation indicated. 2. A monomial is an algebraic expression whose last operation is neither addition not subtraction. 3. So, a monomial is either a separate number represented by a letter or by a figure, for example -x, +9, or a product, for example ab, (x+y), or a quotient, for example , or a power x3, but must never be either a sum or a difference. 4. An algebraic expression which consists of several monomials connected by the plus and minus signs, is known as a polynomial. 5. Such is, for instance, the expression x + yz + c-3 + 6. Terms of a polynomial are separate expressions which form the polynomial by the aid of the + and - signs. 7. Usually the terms of a polynomial are taken with the signs preceding them; for instance, we say: term -a, term +b3 and so on. 8. When there is no sign before the first term it is xy or + xy.
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