Factors, coefficients and combining terms

Factors. If two or more numbers (arithmetic or literal) are multiplied the result of the multiplication is called a product. Each number that has been multiplied to arrive at¹ that product is called a factor of the product. For example, since 2×7=14, the 2 and 7 are factors of their products, 14. Similarly, the number 210 can be written as 2×3×5×7. Two, three, five and seven are called the prime factors of 210. A prime factor is a factor that is not divisible by anything other than² itself or unity. One factors³ 6 when he writes it in the form 2×3. 30 has 2, 3 and 5 as factors. Consider the number 6ab, which can be written as 2×3×a×b. Then 6ab has the following factors: 2, 3ab, a, b, 6 and so on.

When a product is broken down into its factors, it is broken down into numbers which, multiplied together, will equal the product. 2 and 2 are factors of 4. 2 and x are factors of 2x.

Coefficients. Any factor of a product may be called the coefficient of the product of the remaining factors. For example, in the expression 7xyz, 7 is the coefficient of the remaining factors xyz, or 7x is coefficient of the remaining factors yz, etc.

A coefficient which is an arithmetic number is called a numerical coefficient. Thus, 8 is the numerical coefficient in the expression 8xy. If a letter is written without a number before it, the coefficient is understood to be 1. For example, x means 1x, and ab means 1ab.

Combining Terms. An algebraic expression consists of one or more terms. If an algebraic expression consists of more than one term, as for example, 3a 2b c, the terms are separated by plus (+) or minus (—) signs.

A term or a monomial consists of numbers connected only by signs of multiplication or division. For example, 2xy and ab are terms or monomials. Thus, the algebraic expression 3x-2ab+4 has 3 terms: 3x, 2ab and 4.

The purpose of adding or subtracting numbers or objects is to find out⁴ how many of the same kind we have.

The sum of 3ab and 7ab is 10ab, because 3ab's and 7ab's more like them would yield 10ab's. However, 2a and 3b cannot be added because these are unlike terms.

Like terms have the same literal factors. Thus, 3a and 5a are like terms, and xy and 4xy are like terms. Unlike terms do not have the same literal factors. 3d, 7x, 2y and 5xy are all unlike terms.

Adding and Subtracting Like Terms. An algebraic expression containing two or more terms can be simplified by combining like terms. Since unlike terms cannot be added or subtracted we merely indicate their addition or subtraction by signs. For example, 3x+6a-2b.

Notes:

¹ to arrive at — щоб отримати

² is not divisible by anything other than — не ділится ні на що інше, крім

³ to factor — розкладати на множники

⁴ is to find out — полягяє в тому, щоб дізнатися

EXERCISES

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

combining, containing, remaining, following, addition, expression, multiplication, subtraction.

II. Form nouns and translate them into Russian:

to arrive, to multiply, to add, divisible, to consider, to subtract, to express.

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

anything other than, to be broken into numbers, to arrive at, remaining factors, cannot be added, unlike terms, to be simplified.

IV. Answer the following questions:

1. What is the result of multiplication called? 2. What numbers are called factors? 3. What coefficient is called a numerical coefficient? 4. When is the coefficient considered to be 1? 5. By what are the terms separated? 6. What is the purpose of adding or subtracting number? 7. What do like terms have? 8. How do we simplify an algebraic expression?

V. Translate into Ukrainian:

Coefficient is a number or letter or symbol which has a fixed value; generally placed in front of a mathematical expression of letters or symbols and used as a multiplier. Common factor is a number, quantity or expression that divides exactly into two or more numbers, quantities or expressions.

VI. Translate into English:

Одночлен являє собою добуток числового множника і букв. Числовий множник, який стоїть перед буквенними множниками, називаєтся коефіцієнтом. Так, наприклад, число 4 є коефіциієнтом одночлена 4а²ху. Якщо добуток містить декілька числових множників, то зазвичай перемножують їх окремо і отриманий добуток ставлять перед буквенними співмножниками.

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THE FORMULA

The selling price of an article is equal to the sum of the cost of the article and the gain on the cost. If we let S. P. stand for the selling price¹, C for the cost and G for the gain, then a rule for finding the selling price can be written as the equation S.P.=C-G.

Often we have to write² the rule which is being expressed³ briefly as a formula. State the rule for the formula r = d/t. If r is the average rate, d is the distance, and t is the time, then the average rate is equal to the quotient of the distance divided by the time. Or, the average rate is equal to the distance divided by the time.

If the value of every letter but one in a formula is known, the value of the unknown letter can be found. This is known as calculating a formula for the unknown. To evaluate a formula we substitute numerical values for literal numbers, and solve the problem.

Notes:

¹ if we let S.P. stand for the selling price — якщо ми позначимо ціну продажу літерами S. P.

² we have to write — ми повинні написати

³ is being expressed — виражається

EXERCISES

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

rule, evaluation, calculation, expression, equal, quantity.

II. Write these words in the ing-form:

to sell, to be, to find, to gain, to write, to divide, to state, to calculate, to substitute, to provide, to call, to understand, to know, to determine, to value, to take, to vary

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

to stand for, the average rate, to calculate a formula, to solve the problem, to evaluate a formula.

IV. Translate into Ukrainian:

Formula is a general expression for solving certain problems or cases. It is a relation established amongst quantities any one of which may be taken as the unknown if the other quantities are known or can be ascertained. In finding formulas, we are usually given the values of the related numbers to determine how each one relates to or depends upon the other. The related numbers are called variables, for their values vary. When we understand how the numbers vary, we can express in a formula the relationship between the variables.

V. Translate into English:

Часто нам доводиться писати правила, які виражаються коротко у вигляді формули. Якщо значення всіх букв у формулі, за винятком однієї (but one), відоме, то можна знайти значення невідомої букви. Щоб обчислити формулу, ми замінюємо буквенні вирази чисельними і вирішуємо задачу.

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