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.

Ex. 17. Translate into English.

1. Алгебра – это система правил, касающихся действий с числами. 2. В алгебре числа обозначаются буквами, а не цифрами. 3. Поскольку буквы обозначают числа, все законы арифметики годны для действий с буквами. 4. Знаки, которые обозначают действия с цифрами, также употребляются для букв. 5. Операции сложения, вычитания, умножения, деления, возведения в степень и извлечения корней называются алгеб­раическими операциями. 6. В алгебре мы применяем следующие знаки: плюс, минус, знак равенства, знак умножения, знак деления, скобки круглые, квадратные и фигурные, знак «больше, чем», знак «меньше, чем» и другие. 7. Алгебраическому выражению можно придать более простую форму путём приведения подобных членов. 8. Алгебраическая сумма нескольких одночленов называется многочленом. 9. Двучлен – это алгебраическое выражение, состоящее из двух членов, трёхчлен – алгебраическое выражение, состоящее из трёх членов.

Unit 7

Grammar: Present, Past, Future Indefinite, Passive Voice

P resent Indefinite Passive	P ast Indefinite Passive	Future Indefinite P assive
I am He/She/It is asked We/You/They are	I/He/She/It was We/You/They were asked	I/We shall/will be He/She/It will be asked You/They will be
Universal truths, repeated actions indicated by adverbials of frequency usually, always, often, sometimes, seldom, rarely, never, as a rule, every (day, month, etc.), once (a week, a month, etc.).	A single action or a state with time adverbials such as yesterday, ago, last (time, week, month, year), the other day, in 2008.	A predicted future action, an action which the speaker regards as possible to happen in future with the adverbials of time tomorrow, the day after tomorrow, in (a week, month, year), next (week, month, etc.), in 2015.
1. She is often asked at the lessons. – Ее часто спрашивают на уроках. 2. Computers are used everywhere. – Компьютеры используются везде.	1. He was asked at the lesson yesterday. – Его спросили на уроке вчера. 2. A computer was used to solve a difficult problem last week. – Компьютер был использован для решения трудной задачи на прошлой неделе.	1. You will be asked at the lesson tomorrow. – Вас спросят на уроке завтра. 2. This new computer will be used in our laboratory next week. – Этот новый компьютер будет использоваться в нашей лаборатории на следующей неделе.

Ex. 1. Read these sentences. Compare the predicates in these pairs of sentences

a

I attend all lectures at the University. We often use the Internet. He applies a new method in his research. Scientists know all fundamental discoveries.

All lectures at the University are attended by me. The Internet is often used by us. A new method is applied in his research. Новый метод применяется в его исследовании. All fundamental discoveries are known to our scientists.

b	He finished the article yesterday. We studied the rule at the lesson last week. I developed a new system of notation.	The article was finished by him yesterday. The rule was studied at the lesson last week. Это правило изучили на уроке на прошлой неделе. A new system of notation was developed by me.
с	We shall discuss a new program at the lecture tomorrow. He will answer all your questions after the lecture. The writer will publish his book next year.	A new program will be discussed at the lecture tomorrow. All your questions will be answered after the lecture. The book will be published by the writer next year. Книга будет опубликована писателем в следующем году.
d Interrogative	The operations with symbols are performed in algebra. The first coefficient is represented by the letter. Some interesting facts were found in that book. The report will be prepared in time.	Are the operations with symbols performed in algebra? Is the first coefficient represented by the letter? Were any interesting facts found in that book? Will the report be prepared in time?
e Negative	This combination is used in the new system. Such members are easily multiplied. I was told to solve another equation. The order of the operations will be discussed later.	This combination is not used in the new system. Such members are not easily multiplied. I was not told to solve another equation. The order of the operations will not be discussed later.

Ex. 2. State the voice of the verb in the following sentences. Translate these sentences.

1. The students left the experiment unfinished. 2. Algebraic language is used to express mathematical ideas. 3. The members of the equality are connected by the equality sign. 4. The result will be checked immediately. 5. We shall study higher mathematics next term. 6. This property was discussed in the previous chapter. 7. All the facts are summarized in this statement. 8. Will the test be written on Monday? 9. The student showed me his graduation paper a few days ago. 10. She will be told about their recent investigations in the field of algebra. 11. They told the foreign scientists about their studies in the theory of programming. 12. Their calculations will not be used in his work.

Pre-Reading Activity

Guess the meaning of the following words.

Expression [Iks`preS(q)n], identical [aI`dentik(q)l], conditional [kqn`dIS(q)nl], accuracy [`xkjurqsI], classify [`klxsifaI], linear [`lInIq], transformation [,trxnsfq`meiS(q)n], original [q`rIGqnl], reduce [rI`djHs].

Read and learn the basic vocabulary terms.

equation (n) [i`kweiSqn] – уравнение

statement (n) [`steItmqnt] – утверждение, формулировка

finite (a) [`faInaIt] – конечный

variable (a) [`vFqriqbl] – переменная

identical (a) [aI`dentIkql] – аналогичный

briefly (adv) [`brJflI] – кратко

root (n) [rHt] – корень

aid (n) [eId] – помощь

illustrate (v) [`IlqstreIt] – иллюстрировать

restriction (n) [rIs`trIkSqn] – ограничение

substitute (v) [`sAbstitjHt] – замещать, заменять

satisfy (v) [`sxtIsSaI] – удовлетворять

linear (a) [`lInIq] – линейный

quadratic (a) [kwq`drxtIk] – квадратичный

cubic (a) [`kjHbIk] – кубический

integral (n) [`IntIgrql] – интеграл, целое число

fractional (a) [`frxkSqnql] – дробный

rational (a) [`rxSqnl] – 1. рациональный; 2. целесообразный

irrational (n) [i`rxSqnl] – иррациональное число

original (a) [O`rIGqnl] – первоначальный

extraneous (a) [eks`treInjqs] – посторонний, чуждый

Memorize the following word combinations.

1. other than – кроме;

2. in question – рассматриваемый;

3. to check the accuracy – проверить точность;

4. for convenience – для удобства;

5. regardless of the form – независимо от формы;

6. when applied to an equation – в применении к уравнению;

7. is said to be equivalent with respect to – считается эквивалентным относительно;

8. can readily be recognized and discarded – можно легко распознать и отбросить.