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Ex.1. Analyze these pairs of sentences and compare the predicates given there.

1. They are solving the equation at the moment. 2. He was dividing these numerals at 2 o’clock yesterday. 3. We have already reduced the fraction. 4. When I came back you had already replaced the terms in the equation. 5. They will have discussed the definition by 3 o’clock. The equation is being solved at the moment. These numerals were being divided at 2 o’clock yesterday. The fraction has already been reduced. When I came back the terms in the equation had already been replaced. The definition will have been discussed by 3 o’clock.
1. Are you changing the improper fraction to the whole number now? 2. Was she writing the decimal fractions at that moment? 3. Has he omitted the plus sign in this sentence? 4. Will you have translated the article by tomorrow? Is the improper fraction being changed to the whole number now? Were the decimal fractions being written at that moment? Has the plus sign been omitted in this sentence? Will the article have been translated by tomorrow?
1. The students are not multiplying these integers right now. 2. We were not subtracting the fractions when the teacher came in. 3. You have not divided the numerator yet. 4. The student had not proved the theorem by the end of the class yesterday. 5. They will not have checked the result of the calculation by 5 o’clock tomorrow. These integers are not being multiplied by the students right now. The fractions were not being subtracted when the teacher came in. The numerator has not been divided yet. The theorem had not been proved by the end of the class yesterday. The result of the calculation will not have been checked by 5 o’clock tomorrow.

 

Ex. 2. State the voice and the tense-form of the verbs in the following sentences and translate them into Russian.

1. The students are being given a lecture now. 2. The students were being asked about mathematical sentences the whole lesson. 3. The given quantity hasn’t been divided yet. 4. All the data had been obtained by that time. 5. The algorithm will have been carefully worked out by tomorrow. 6. Have any of these articles on mathematics been translated recently? 7. All the digits have already been aligned as appropriate. 8. The conference is being held at the moment. 9. Are the numbers being added without a calculator right now? 10. His graduation paper hasn’t been presented yet.

 

 

Ex. 3. Open the parentheses and give the correct form of the verb in the Passive Voice.

1. Don’t enter the room! A student (to examine) there just now. 2. The letter (to type) by the typist when I came in. 3. I am sure that his work (to complete) by the end of the month. 4. A lot of new words (to learn) already by the students. 5. All the dinner (to eat) before they finished the conversation. 6. The question (not to answer) yet. 7. The proposal (to consider) by 9 o’clock yesterday. 8. The papers (to sign) just by the dean of the faculty. 9. The results of the test (to discuss) by the students at the moment. 10. The article (to translate) by the time you return.

 

Pre-Reading Activity

Guess the meaning of the following words.

Algebraic adj. ["xlGI`breIIk]; polynomial n. ["pOlI`nqumIql]; integral adj. [`IntIgr(q)l]; constant n. [`kOnstqnt]; coefficient n. ["kquI`fISqnt]; exponent n. [eks`pqunqnt]; fundamental adj. ["fAndq`mentl]; process n. [`prquses].

 

Read and learn the basic vocabulary terms:

term n. [tE:m] член, термин;
upper adj. [`Apq] верхний, высший;
degree n. [dI`grJ] степень;
appear v. [q`pIq] показываться, появляться;
above prep. [q`bAv] над, выше, сверх;
monomial n. [mO`nqumIql] одночлен, adj. – одночленный;
binomial n. [baI`nqumIql] двухчлен, adj. – двухчленный;
trinomial n. [traI`nqumIql] трёхчлен, adj. – трёхчленный;
power n. [`pauq] показатель степени;
place v. [pleIs] размещать, ставить;
obtain v. [qb`teIn] получать, приобретать;
arrange v. [q`reInG] размещать, располагать, расставлять;
arrangement n. [q`reInGmqnt] размещение, расположение;
ascend v. [q`send] подниматься, восходить;
descend v. [dI`send] спускаться, снижаться;
state v. [steIt] сообщать, формулировать;
concern v. [kqn`sE:n] касаться, иметь отношение.
precede v. [prI`sJd] предшествовать, стоять перед чем-л.

Memorize the following word combinations.

1. such as – такой как

2. in other words – другими словами

3. is known as a polynomial – известен как многочлен, называется многочленом

4. thus – таким образом

5. either … or – либо … либо, или … или

6. in order to – для того, чтобы

7. by the aid of – с помощью (чего-либо)

Reading Activity

Polynomials

A number represented by algebraic symbols is referred to as an algebraic expression.

An algebraic expression whose parts are not separated by + or is called a term; as 2×3, –5×yz, and xy/z.

In the expression 2×3 – xyz – xy/z there are three terms. The expression c×(a-b) is a term.

An algebraic expression of one term is known as a monomial or a simple expression. (xy and 3ab are monomials).

An algebraic expression of more than one term is called a polynomial. Such is, for instance, the expression ab – a + b – 10 + (a – b)/c.

In other words we can say that algebraic expressions which consist of several monomials connected by the + and signs are known as polynomials.

Terms of a polynomial are separate expressions which form the polynomial by the aid of addition and subtraction. Usually, the terms of a polynomial are taken with the signs preceding them; for instance, we say: term –a, term +,and so on.

A polynomial of two terms is called a binomial, e.g. 3a+2b and x² – y² are binomials. Similarly a+b+c is a trinomial.

Thus, all algebraic expressions are divided into two groups according to the last algebraic operation indicated: monomials and polynomials.

An expression, any term of which is a fraction, is referred to as a fractional expression, as –3x+a/x; all the other expressions are called integral ones.

An algebraic expression such as 5x³–7x² + 9x + 6 is a polynomial or an integral expression in the letter x. It is composed of one or more terms, each of which is either an integral power of x multiplied by a constant or a constant which is free of x. The constant multipliers 5, 7, 9 are called coefficients; the upper numbers: 3, 2 are exponents; 6 is the constant term. The polynomial is of the third degree in x since 3 is the highest exponent appearing in the expression.

An expression such as 2x²y+5x³yz³-9xyz+2x+7 is a polynomial in x, y and z. The degree of a polynomial in several letters is the highest degree that any single term has in those letters. Thus, the above expression is of the seventh degree in x, y and z since the sum of the exponents of the second term is seven.

Let’s consider four fundamental operations of polynomials.

The first operation is addition. In order to add polynomials, you should place them in such a way that like terms fall under each other, and add the coefficients in each column to find the final coefficient of that term.

The second one is subtraction. To subtract one polynomial from another place the terms of the subtrahend under like terms of the minuend, change the signs of the terms of the subtrahend and add.

The third operation is multiplication. Suppose, you have been given two polynomials and have been asked to multiply one of them by the other. In order to do it, you are to multiply each term of one by every term of the other and to add the products thus obtained.

And the last one is division. To divide one polynomial by another, arrange both the dividend and the divisor in ascending or descending powers of some letter common to both and write the quotient as a fraction.

The rule concerning the operation of division may be stated in the following way:

1. Divide the leading term of the dividend by the leading term of the divisor, obtaining the first term of the quotient.

2. Multiply each term of the divisor by this term of the quotient and subtract the product from the dividend.

The remainder found by this subtraction is used as the dividend and the process is repeated. The work is continued until a remainder is reached which is of lower degree than the divisor. In any case of division if the remainder is zero, the division is exact.

Post-Reading Activity

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