# Factor

# Factor

In mathematics, to factor a number or algebraic expression is to find parts whose product is the original number or expression. For instance, 12 can be factored into the product 6 × 2, or 3 × 4. The expression (x^{2}-4) can be factored into the product (x+2) (x-2). Factor is also the name given to the parts. Mathematicians say that 2 and 6 are factors of 12, and (x-2) is a factor of (x^{2}-4). Thus, mathematicians refer to the factors of a product and the product of factors.

The fundamental theorem of arithmetic states that every positive integer can be expressed as the product of prime factors in essentially a single way. A prime number is a number whose only factors are itself and 1, where the first few prime numbers are 1,2, 3, 5, 7, 11, and 13. Integers that are not prime are called composite. The number 99 is composite because it can be factored into the product 9×11. It can be factored further by noting that 9 is the product 3×3. Thus, 99 can be factored into the product 3×3×11, all of which are prime. By saying ‘in essentially one way,’ it is meant that although the factors of 99 could be arranged into 3×11×3 or 11×3×3, there is no factoring of 99 that includes any primes other than 3 used twice and 11.

Factoring large numbers was once mainly of interest to mathematicians, but today factoring is the basis of the security codes used by computers in military codes and in protecting financial transactions. High-powered computers can factor numbers with 50 digits, so these codes must be based on numbers with a hundred or more digits to keep the data secure.

In algebra, it is often useful to factor polynomial expressions (expressions of the type 9x^{3}+3x^{2} or x^{4}-27xy+32). For example, x^{2}+4x+4 is a polynomial that can be factored into (x+2)(x+2). That this is true can be verified by multiplying the factors together. The degree of a polynomial is equal to the largest exponent that appears in it. Every polynomial of degree n has at most n polynomial factors (though some may contain complex numbers). For example, the third degree polynomial x^{3}+6x^{2}+11x + 6 can be factored into (x+3) (x^{2}+3x+2), and the second factor can be factored again into (x+2)(x+1), so that the original polynomial has three factors. This is a form of (or corollary to) the fundamental theorem of algebra.

In general, factoring can be rather difficult. There are some special cases and helpful hints, though, which often make the job easier. For instance, a common factor in each term is immediately factorable; certain common situations occur often and one learns to recognize them, such as x^{3} + 3x^{2}+xy = x(x^{2}+3x+y). The difference of two squares is a good example: a^{2}-b^{2} = (a+b)(a-b). Another common pattern consists of perfect squares of binomial expressions, such as (x + b)^{2}. Any squared binomial has the form x^{2}+2bx+b^{2}. The important things to note are: (1) the coefficient of x^{2} is always one (2) the coefficient of x in the middle term is always twice the square root of the last term. Thus, x^{2}+10x+25 = (x+5)^{2}, x^{2}-6x+9 = (x-3)^{2}, and so on.

Many practical problems of interest involve polynomial equations. A polynomial equation of the form ax^{2}+bx+c = 0 can be solved if the polynomial can be factored. For instance, the equation x^{2}+x-2 = 0 can be written (x+2)(x-1) = 0, by factoring the polynomial. Whenever the product of two numbers or expressions is zero, one or the other must be zero. Thus, either x+2 = 0 or x-1 = 0, meaning that x = -2 and x = 1 are solutions of the equation.

## Resources

### BOOKS

Bittinger, Marvin L. *Basic Mathematics*. Boston, MA: Addison-Wesley Publishing, 2003.

### KEY TERMS

**Product—** The product of two numbers is the number obtained by multiplying them together.

Bittinger, Marvin L, and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications*. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Jeffrey, Alan. *Mathematics for Engineers and Scientists*. Boca Raton, FL: Chapman & Hall/CRC, 2005.

Lorenz, Falko. *Algebra*. New York: Springer, 2006.

Setek, William M. *Fundamentals of Mathematics*. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

J.R. Maddocks

# Factor

# Factor

In **mathematics** , to factor a number or algebraic expression is to find parts whose product is the original number or expression. For instance, 12 can be factored into the product 6 × 2, or 3 × 4. The expression (x2 - 4) can be factored into the product (x + 2)(x - 2). Factor is also the name given to the parts. We say that 2 and 6 are factors of 12, and (x-2) is a factor of (x2 - 4). Thus we refer to the factors of a product and the product of factors.

The fundamental **theorem** of **arithmetic** states that every positive integer can be expressed as the product of prime factors in essentially a single way. A prime number is a number whose only factors are itself and 1 (the first few **prime numbers** are 1, 2, 3, 5, 7, 11, 13). **Integers** that are not prime are called composite. The number 99 is composite because it can be factored into the product 9 × 11. It can be factored further by noting that 9 is the product 3 × 3. Thus, 99 can be factored into the product 3 × 3 × 11, all of which are prime. By saying "in essentially one way," it is meant that although the factors of 99 could be arranged into 3 × 11 × 3 or 11 × 3 × 3, there is no factoring of 99 that includes any primes other than 3 used twice and 11.

Factoring large numbers was once mainly of interest to mathematicians, but today factoring is the basis of the security codes used by computers in military codes and in protecting financial transactions. High-powered computers can factor numbers with 50 digits, so these codes must be based on numbers with a hundred or more digits to keep the data secure.

In **algebra** , it is often useful to factor polynomial expressions (expressions of the type 9x3 + 3x2 or x4 27xy + 32). For example x2 + 4x + 4 is a polynomial that can be factored into (x + 2)(x + 2). That this is true can be verified by multiplying the factors together. The **degree** of a polynomial is equal to the largest **exponent** that appears in it. Every polynomial of degree n has at most n polynomial factors (though some may contain **complex numbers** ). For example, the third degree polynomial x3 + 6x2 + 11x + 6 can be factored into (x + 3) (x2 + 3x + 2), and the second factor can be factored again into (x + 2)(x + 1), so that the original polynomial has three factors. This is a form of (or corollary to) the fundamental theorem of algebra.

In general, factoring can be rather difficult. There are some special cases and helpful hints, though, that often make the job easier. For instance, a common factor in each **term** is immediately factorable; certain common situations occur often and one learns to recognize them, such as x3 + 3x2 + xy = x(x2+ 3x + y). The difference of two squares is a good example: a2 - b2 = (a + b)(a - b). Another common pattern consists of perfect squares of binomial expressions, such as (x + b)2. Any squared binomial has the form x2 + 2bx + b2. The important things to note are: (1) the **coefficient** of x2 is always one (2) the coefficient of x in the middle term is always twice the **square root** of the last term. Thus x2 + 10x + 25 = (x+5)2, x2 - 6x + 9 = (x-3)2, and so on.

Many practical problems of interest involve polynomial equations. A polynomial equation of the form ax2 + bx + c = 0 can be solved if the polynomial can be factored. For instance, the equation x2 + x - 2 = 0 can be written (x + 2)(x - 1) = 0, by factoring the polynomial. Whenever the product of two numbers or expressions is **zero** , one or the other must be zero. Thus either x + 2 = 0 or x - 1 = 0, meaning that x = -2 and x = 1 are solutions of the equation.

## Resources

### books

Bittinger, Marvin L, and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications.* 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Davison, David M., Marsha Landau, Leah McCracken, Linda Immergut, and Brita and Jean Burr Smith. *Arithmetic and Algebra Again.* New York: McGraw Hill, 1994.

Larson, Ron. *Precalculus.* 5th ed. New York: Houghton Mifflin College, 2000.

McKeague, Charles P. *Intermediate Algebra.* 5th ed. Fort Worth: Saunders College Publishing, 1995.

J.R. Maddocks

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Product**—The product of two numbers is the number obtained by multiplying them together.

# factor

fac·tor
/ ˈfaktər/
•
n.
1.
a circumstance, fact, or influence that contributes to a result or outcome:
*his legal problems were not a factor in his decision*

*she worked fast, conscious of the time factor.*∎ Biol. a gene that determines a hereditary characteristic:

*the Rhesus factor.*2. a number or quantity that when multiplied with another produces a given number or expression. ∎ Math. a number or algebraic expression by which another is exactly divisible. 3. Physiol. any of a number of substances in the blood, mostly identified by numerals, which are involved in coagulation. 4. a business agent; a merchant buying and selling on commission. ∎ a company that buys a manufacturer's invoices at a discount and takes responsibility for collecting the payments due on them. ∎ archaic an agent, deputy, or representative. • v. [tr.] 1. Math. express (a number or expression) as a product of factors. 2. sell (one's receivable debts) to a factor. PHRASAL VERBS: factor something in (or out) include (or exclude) something as a relevant element when making a calculation or decision:

*when the psychological costs are factored in, a different picture will emerge.*DERIVATIVES: fac·tor·a·ble adj.

# Factor

# Factor

In the late Middle Ages, merchants in foreign countries often grouped themselves into a community enjoying mutual protection and sometimes special privileges vis-à-vis local authorities. The head of such a foreign merchant group or colony was termed the "factor" and the community the "factory." Bruges, for example, in the fourteenth and fifteenth centuries, was the site of Portuguese and Castilian factories through which much of those nations' trade with northern Europe was channeled.

When the Portuguese began their expansion down the west coast of Africa, they adapted and made use of this system by establishing fortified warehouses and administrative offices through which they dealt with the native merchants. Such factories (*feitorias*) were established at Arguin in 1451 and at São Jorge de Mina in 1481.

Similarly, shortly after the discovery of Brazil in 1500, the merchant group that had leased the trade rights from the king established a factory there (1504) in order to have a place to store the dyewood awaiting shipment to Portugal. While claims have been made for the existence of Portuguese factories at various points along the coast in the period 1502–1534, according to the historian Rolando Laguardia Trías there was only one indisputable factory in Brazil before the arrival of Martim Afonso de Sousa, and until recently it was thought to have been located at Cabo Frio. Trías, however, gives good reasons for thinking that it was actually situated on an island in Guanabara Bay (Rio de Janeiro). In 1516, with expiration of the royal lease and the crown's resumption of direct control over Brazilian trade, this factory was moved north to Itamaracá near the present-day town of Igaraçu, both because better quality brazilwood was found there and because the site was closer by ship to Lisbon.

*See also***Brazilwood; Portuguese Overseas Administration.**

## BIBLIOGRAPHY

*História naval Brasileira*, vol. 1 (1975), pp. 254-256.

John Vogt, *Portuguese Rule on the Gold Coast, 1469–1682* (1979).

A. H. De Oliveira Marques, *Ensaios de história medieval Portuguesa*, 2d ed. (1980), pp. 164-166, 178-179.

**Additional Bibliography**

Bueno, Eduardo. *Náufragos, traficantes e degredados: As primeiras expedições ao Brasil, 1500–1531*. Rio de Janeiro: Objetiva, 1998.

Machado, Paulo Pinheiro. *A política de colonização do Império*. Porto Alegre, RS: Editora da Universidade, Universidade Federal do Rio Grande do Sul, 1999.

Harold B. Johnson

# factor

**factor** **1.** (stat.) One of a pair or series of numbers which when multiplied together yield a given product.

**2.** (**limiting factor**) See LIMITING FACTOR.

**3.** (**ecological factor**) See LIMITING FACTOR.

# factor

# factor

**factor** **1.** In statistics, one of a pair or series of numbers which yield a given product when multiplied together.**2.** (*ecological factor*) See limiting factor.

# Factor

# FACTOR

*An event, circumstance, influence, or element that plays a part in bringing about a result.*

A factor in a case contributes to its causation or outcome. In the area of negligence law, the *factors*, or *chain of causation*, are important in determining whether liability ensues from a particular action done by the defendant.

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**factor**