Polynomials - раздел Образование, Grammar: verb “to be”, there is/are. Possessive and Personal Pronouns A Number Represented By Algebraic Symbols Is Referred To As An Algebraic Expr...
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 +b², 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.
Все темы данного раздела:
Ex. 1. Analyze the following sentences and translate them into Russian.
1. They live together with their parents. 2. Oxford is famous for its University. 3. I didn’t have an
C) Preparatory “it”.
It
is
was
will be
easy/difficult/impossible/dangerous/safe/
expensive/interesting/nice/wonderful/
terrible/a pleasure
Ex. 4. Make the following sentences interrogative and negative.
a) 1. Twenty miles is a long way to walk. 2. My native city is very large. 3. The pair of black trousers is cheap.4. Phonetics is a branch of linguistics. 5. The family are fond of their house. 6.
Ex. 6. Analyze the following sentences and translate them into Russian.
a) 1. There are some big trees in the garden. 2. There is a seminar on philosophy today. 3. There are comfortable apartments in this block of flats. 4. There are a lot of accidents on this road. 5.
Read and learn the basic vocabulary terms.
sense (n) [sens]
ощущение, чувство
support (v, n) [sq'pLt]
содержать (семью); поддержка
seem (v)
About My Family and Myself
I believe that everything has its beginning in the family. Family is very important for every person, because it gives you a sense of stability and tradition, a feeling of having support and unders
Looks and Appearance
beautiful (adj) ['bjHtIful]
красивый (о женщинах)
blond/fair [feq]/ginger [GInGq]/ dark hair
светлые/русые/рыжие/черные в
Features of Character
be in good/bad mood [mu:d]
быть в хорошем (плохом) настроении
brave (adj) [breIv]
смелый
devoted
Interests and Ambitions
ambition (n) [xm'bISqn]
стремление
be keen [kJn] on smth
увлекаться чем-либо
desire (n), (v) [di'
Family Members and Relations in the Family
average ['xvqrIG] /small/large family
средняя/маленькая/большая
семья
consist of (v) [kqn'sIst]
состоят
Ex. 12. Translate into English making use of the words from the box.
slender
beard
shy
short
fat
hazel eyes
pale skin
kind-hearted
plump cheeks
honest
ginger-haired
curly hair
broad shoulders
Ex. 1. Make the sentences interrogative and negative.
a) 1. I usually have a sandwich for my lunch. 2. Students have one English class a week. 3. We have a shop next to the post office. 4. Most cars have got four wheels. 5. An insect has got six legs.
Present Simple
Positive
Negative
Interrogative
I
we
you
they
Ex. 4 . Make the following sentences interrogative and negative.
1. We do a lot of things in our free time. 2. The shops open at 9 o’clock and close at 5.30. 3. It costs a lot of money to stay in luxury hotels. 4. The Moon goes round the Earth. 5. They usually s
Past Simple
Positive
Negative
Interrogative
I
we
you
they
he
she
Future Simple
Positive / Negative
Interrogative
I/we
you/they
he/she/it
(shall(‘ll)/shall not/shan’t) will(‘ll)/wil
General
Auxiliary verb
Subject
Predicate or part of it
Obje
Read and learn the basic vocabulary terms.
delay (n) [dI`leI]
задержка
recent (adj) [`rJs(q)nt]
последний, недавний
manage (v) [`mxnIG]
Memorise the following word combinations.
to be busy
быть занятым
the students’ hall of residence
студенческое общежитие
as a matter of fac
A Letter to a Friend
Dear Linda,
I’m very sorry for the delay in answering your recent letter. I was so awfully busy. In the spring I passed my A-Level examinations and tests at school and managed to get good
Ex. 10. Find the Russian equivalents for the following English word combinations.
1. to belong to a group
2. to describe the routine
3. to depend on the weather
4. to have a late night
5. on the other hand
Pronouns some, any, no
Affirmative sentences
Examples
Some and its compounds (somebody, something, somewhere, someone) is used with the meaning “small amo
Read and learn the basic vocabulary terms.
number (n) [`nAmbq]- число, количество, номер
date back to (v) [`deIt] – датироваться, относиться к определенному времени
antiquity (n) [xn`tIkwItI]- древность, античность
Numbers
The beginning of our number system dates back to antiquity where symbols, which we call positive integers, were used as an aid in counting, and only in the nineteenth century the system, which we k
Ex. 8. Answer the following questions.
1. What were positive integers used for? 2. When was the number system completed? 3. Are the first numbers which we use the positive integers or the negative ones? 4. Is unity a first or a last int
Ex. 1. Analyze the following sentences and translate them into Russian.
1. She could do sums in her head when she was 6 years old. 2. Peter was able to carry out the experiment successfully, but Nick couldn’t finish it without any help.
Read and learn the basic vocabulary terms.
number (n) [`nAmbq] число, количество; v перечислять
numeral (n) [`nju:mqrql] цифра, символ, число
digit (n) [`dIGIt] цифра
value (n) [`vxlju:] величина, значение; (v) це
Four Basic Operations of Arithmetic
Many thousands of years ago this was a world without numbers. Today, using the same numbers in many different ways, man can build bridges, skyscrapers, fly off the Earth like a bird, even measure t
Ex. 14. Choose the phrase closest in meaning to the given statement.
1. Dan can’t be a teacher.
a) I’m sure Dan isn’t a teacher.
b) I think Dan isn’t a teacher.
2. Need I take the tablets every day?
a) Is it a good idea to take th
Ex. 4. Translate the following sentences into Russian.
1. Mathematicians have used mathematical formulas in solving these problems. 2. By the end of the lesson we’ll have been able to obtain the modified definition of the function. 3. Scientific theori
Read and learn the basic vocabulary terms.
part [pa:t] n
часть
proper ['prOpq] a
правильный
improper [im'prOpq] a
неправильный
Rational numbers and decimal numerals
In mathematics, a fraction is a concept of a proportional relation between a part and a whole. In other words, it is an example of a specific type of ratio, in which the two number
Ex.7. Give the proper English equivalents for the Russian expressions.
the greatest common factor; to invert; the quotient; performed operations; the numerator and denominator; decimal numerals; proper fractions; the minuend, subtrahend and r
Ex. 10. Choose the correct tense form.
1. (Have you ever heard/ Did you ever hear) about continued fractions?
2. We (have started/ started) learning fractions when we were at school.
3. By the end of the lecture the st
Ex. 11. Ask special questions.
1. By 5 o’clock the experiment will have been over. (by what time) 2. By the age of 41 Sophia Kovalevskaya had won recognition of mathematicians all over the world. (whose) 3. Zero concept has got
Ex. 12. Translate into English.
1. Дробь представляет часть целого. Она показывает, что что-то разделили на несколько равных частей. 2. В дроби число, стоящее над чертой называется числителем, число, стоящее под чертой называетс
Degrees of Comparison
short words:
-er, -est
big – bigger – the biggest
thin – thinner – the thinnest
short – shorter – the shortest
Ex. 1. Analyze these sentences and compare the adjectives given there. Translate them into Russian.
1. He has a difficult test. I have a more difficult test. Her test is the most difficult of all. 2. Your problem is easy. His problem is easier than yours. That student’s problem is the easiest. 3.
Types of Comparisons
There are a number of different sentence patterns with comparative and superlative forms:
Than чем
This book is more interesting
Ex. 5. Give the proper English equivalents for the Russian expressions.
the best, greater, as difficult as, less, longer… that one, as interesting as, not so important as, the youngest, better, not so famous as, the sooner… the better, the old
Facts to be remembered
a. Present Perfect Continuous is used:
For actions started and finished in the past and lasted for some time. The result of the actions is visible
Note the difference in translation between the Present Perfect Continuous and the Present Perfect Tenses.
1. – You look hot. – I’ve been running all the way.
Я бежала всю дорогу.
2. I’ve been learning irregular verbs all afternoon.
Я учил неправильные глаголы ве
Read and learn the basic vocabulary terms.
compute [kqm`pjHt]
вычислять
deal (dealt) with [dJl]
иметь дело с; рассматривать
apply [q`plaI]
The Nature of Algebra
Algebra is a generalization of arithmetic. Each statement of arithmetic has been dealing with particular numbers for years: the statement (20 + 4)2 = 202 + 2 • 20 • 4 + 4
Ex. 13. Give the proper English equivalents for the Russian expressions.
computed, instead of, simpler, ordinary letters, replace, hold, generalization, relations, concerning, multinomials
1. Algebra is обобщение
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.
Ex. 1. Read these sentences. Compare the predicates in these pairs of sentences
a
1. I attend all lectures at the University.
2. We often use the Internet.
3. He applies a new method in his research.
&n
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
Equations and Identities
An equation is a statement of equality between two algebraic expressions. The two expressions are called members, or sides of the equation. If the two members of an equation are finite and are exac
Ex. 5. Give the proper English equivalents for the Russian expressions.
to solve, the unknown quantity, substituted into the expression, the value, roots, have been transposed, the Least Common Denominator, is checked, clearing, to satisfy
Ex. 9. Translate the sentences from English into Russian.
1. In solving problems by means of algebraic equations, the first and the most difficult step which must be done is to translate the words into the algebraic language. 2. Definite rules cannot be g
Ex. 10. Translate the sentences from Russian into English.
1. Уравнение – это утверждение, выражающее равенство двух алгебраических выражений. 2. Корень уравнения остаётся прежним, если к обеим частям уравнения прибавить или от обеих частей уравнения вычес
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 fr
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]; fundamen
Ex. 6. Give the proper English equivalents for the Russian expressions.
a trinomial, descending, subtraction, obtain, a fraction, coefficients, terms, exponents, sum, remainder, fractional
1. Each of these polynomi
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