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 7 xyz, 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 yeild 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 — чтобы получить

² is not divisible by anything other than — не делится ни на что другое, кроме

³ to factor — разлагать на множители

⁴ is to find out — состоит в том, чтобы узнать