Exercises

I. Read the following words paying attention to the pronunciation:

subtract, illustrate, result, where, which, parentheses, other, whether, always, way, statement, remain, same, equation.

II. Give the Infinitives of the following words:

is, given, finding, got, saw, known.

III. Make up sentences of your own using the words and expressions given below:

may be true for, in finding, in question, by substituting, to consist in omitting, a linear equation, the least common denominator.

IV. Answer the following questions:

1. What is an equation? 2. What are the expressions on either side of the sign of equality called? 3. What should be done to keep the balance of the equation? 4. How do we check an equation? 5. What operations must one do when solving an equation by the combination of rules?

V. Translate into Russian:

If the same number is added to each side of an equation, the equality of the two sides is not altered. If the same number is subtracted from each side of an equation, the equality of the two sides is not altered. If both sides of an equation are multiplied by the same number, the equality of the two sides is not altered. If both sides of an equation are divided by the same number, the equality of the two sides is not altered.

VI. Translate into English:

Корень уравнения остается прежним, если к обеим частям уравнения прибавить или от обеих частей уравнения вычесть одно и то же выражение. Корень уравнения остается прежним, если обе части уравнения умножить или разделить на одно и то же выражение.

Решить уравнение - значит найти те (those) значения неизвестного, при которых обе части уравнения равны одному и тому же числу (другими словами, все те значения неизвестного, при которых равенство будет верным). Говорят, что эти значения неизвестного удовлетворяют уравнению. Значения неизвестного, которые удовлетворяют уравнению, называются корнями или решениями уравнения.

TEXT

Monomial and polynomial

Algebraic expressions are divided into two groups according to the last algebraic operation indicated.

A monomial is an algebraic expression whose last operation in point of order is neither addition nor¹ subtraction.

Consequently, a monomial is either a separate number represented by a letter or by a figure, e.g. -a, +10, or a product, e.g. ab, (a+b)c, or a quotient, e.g. (a-b)/c, or a power, e.g. b2, but it must never be either a sum or a difference.

If a monomial is a quotient, it is called a fractional monomial; all the other monomials are called integral monomials. Thus, (a-b)/c is a fractional monomial, while (x—y)ab; a(x+y)² are integral monomials.

An algebraic expression which consists of several monomials connected by the + and - signs, is known as a polynomial². Such is for instance, the expression

ab-a+b-10+(a-b)/c.

Terms of a polynomial are separate expressions which form the polynomial by the aid of the + and — signs. Usually, the terms of a polynomial are taken with the signs prefixed to them; for instance, we say: term -a, term +62, and so on. When there is no sign before the first term it is ab or +ab.

A binomial is an algebraic expression of two terms; a trinomial is an expression of three terms and so on.

Notes:

¹ neither ...nor – ни…ни

² is known as a polynomial – известен как многочлен, называется многочленом