Factoring
Overview
Factoring a number means representing the number as the product of prime numbers. Prime numbers are those numbers that cannot be divided by any smaller number to produce a whole number. For instance, 2, 3, 5, 7, 11, and 13 (among many others) cannot be divided without producing a remainder.
Factoring in its simplest form is the ability to recognize a common characteristic or trait in a group of individuals or numbers which can be used to make a general statement that applies to the group as a whole.
Another way to think of factoring is that every individual in the group shares something in particular. For example, whether someone is from France, Germany, or Austria is irrelevant in the statement that they are European, because all three of these countries share the geographic characteristic of being on the continent of Europe. The factor that can be applied to all three individuals in this particular group is that they are all European. The ability to recognize relationships between individual components is fundamental to mathematics. Factoring in mathematics is one of the most basic but important lessons to learn in preparation for further studies of math.
Fundamental Mathematical Concepts and Terms
A number which can be divided by smaller numbers is referred to as a composite number.
Composites can be written as the product of smaller primes. For example, 30 has smaller prime numbers which can be multiplied together to achieve the product of 30. These numbers are as follows: 2 × 3 × 5 = 30. A number is considered to be factored when all of its prime factors are recognized. Factors are multiplied together to yield a specific product.
It is important to understand a few basic principals in factoring before further discussion can continue on how factoring can be applied to real life. One of the most important studies of mathematics is to study how individual entities relate to one another.
In multiplying factors which contain two terms, each term must be multiplied with each term of the second set of terms. For example, in (a+b) (a+b), both the a and b in the first set must be multiplied by the a and b in the second set. The easiest way to accomplish this is by employing the FOIL method. FOIL refers to the order of multiplication: first, outer, inner, and last. First we multiply a by a to yield a2, then the Outer terms of a and b to yield ab, then the Inner terms of b and a to yield another ab, finally we multiply the Last terms of b and b for b2. Putting all of these together, we achieve a2 + 2ab + b2.
Greatest common factor (GCF) refers to two or more integers where the largest integer is a factor of both or all numbers. For example, in 4 and 16, both 2 and 4 are factors that are common to each. However, 4 is greater than 2, so therefore 4 is the greatest common factor. In order to find the greatest common factor, you must first determine whether or not there is a factor that is common to each number. Remember that common factors must divide the two numbers evenly with no remainders. Once a common factor is found, divide both numbers by the common factor and repeat until there are no more common factors. It is then necessary to multiply each common factor together to arrive with the greatest common factor.
Factoring perfect squares is one of the essentials of learning factoring. A perfect square is the square of any whole number. The difference between two perfect squares is the breaking of two perfect squares into their factors. For example a2 − b2 is referred to as the difference between two perfect squares. The variables a and b refer to any number which is a perfect square. In order to factor a2 − b2, we must see that the factors must contain both a and b. If we start with (a − b), and remove this expression from a2 − b2, we will have (a − b) remaining. This would yield a solution of (a − b) (a − b). Using the FOIL method, the product would be a2 − ab − ab + b2, which is a2 − 2ab + b2 which is incorrect due to the presence of a middle term.
Alternatively, if we choose (a + b) and remove both a and b from the original equation, we have: (a + b) (a − b). Multiplying these factors back together yields a2 − ab + ab − b2 which simplifies to our original equation of (a2 − b2). The difference between two perfect squares always has alternating + and − signs to eliminate the middle term.
Real-life Applications
Factoring is used to simplify situations in both math and in real life. They allow faster solutions to some problems. In the mathematical calculations used to model problems and derive solutions, factoring plays a key role in solving the mathematics that describe systems and events.
IDENTIFICATION OF PATTERNS AND BEHAVIORS
By learning the patterns and behaviors of factors in mathematical relationships, it is possible to identify similarities between multiple components. By being able to quickly and accurately find similarities, a solution can usually be identified. The solution to any given problem is based on how each individual player or factor in the problem relates to one another for an effective solution. By being able to see these relationships, many times it is possible to see the solution in the relationship.
An example is commonly found in decision making. For example, a shopper enters an unfamiliar grocery store looking for Gouda cheese. The shopper could wander aimlessly, hoping to spot the cheese, but a smarter approach illustrates the intuitive process of factoring. Granted, with enough time, the shopper might eventually find the cheese, but a better approach is to search for a common factor to help narrow the search. What common factor does cheese have with other items in the store? The obvious choice would be to look for the dairy section and eliminate all other sections in the store. The shopper would then further factor the problem to locate the cheese section and eliminate the milk, eggs, etc. Finally one would only look at the cheese selections for the answer, the Gouda cheese. This is a fairly simple nonmathematical example, but it demonstrates the principle of mathematical factoring—a search for similarities among many individual numerical entities.
REDUCING EQUATIONS
In math, one of the most useful applications of factoring is in eliminating needless calculations and terms from complex equations. This is often referred to as "slimming down the equation." If you can find a factor common to every term in the equation, then it can be eliminated from all calculations. This is because the factor will eventually be eliminated through the calculation and simplification process anyway. An example of this is (2 + 8)/4 which can be slimmed down to (1 + 4)/2 by eliminating the common factor of 2. The value of the first expression was 10/4 and the value of the second one is 5/2, which is the same once 10/4 is simplified. As we can see, one advantage in eliminating factors is the answer is already simplified. Now let's take a look at a slightly more complicated example:
we can see that a common factor of ax2 can be eliminated.
This expression then becomes:
This same technique can be employed in any mathematical equation in which there is a factor common to all parts of the equation.
DISTRIBUTION
Factoring is often used to solve distribution and ordering problems across a range of applications. For example, a simple factoring of 28 yields 4 and 7. In application, 28 units can be subdivided into 4 groups of 7 or 7 groups of 4, Again, by example, in application 28 players could be divided into 4 teams of 7 players or 7 teams of 4 players. This is intuitive factoring—something done every day without realizing that it is a math skill.
SKILL TRANSFER
In addition to factoring mathematical equations, the ability to mathematically factor has been demonstrated to transfer into stronger pattern recognition skills that allow rapid categorization of non-mathematical "factors." Essentially is it an ability to find and eliminate similarities and thus focus on essential difference.
When a defensive linebacker looks over an offensive set in football, he scans for patterns and similarities in numbers of players each side of the ball, in the backfield, in an effort to determine the type of play the opposing quarterback (or his coach) has called. This is not mathematical factoring, but psychology studies have shown that practice in mathematical factoring often leads to a general improvement in pattern recognition and problem solving.
CODES AND CODE BREAKING
Another example of mathematical factoring is in coding and decoding text. Humans have found clever ways of concealing the content of sensitive documents and messages for centuries. Early forms of coding involved the twisting of a piece of cloth over a rod of a certain length. On the cloth would be printed a confusing matrix of seemingly unrelated letters and symbols. When the cloth was twisted over a rod of the proper diameter and length, it would align letters to form messages. The concealed message would be determined by a mathematical factor of proper rod diameter and length that only the intended party would have in possession. Coding and decoding text today is far more complicated. In our new highly computerized age, coding and decoding text depends on an extremely complicated algorithm of mathematical factors.
GEOMETRY AND APPROXIMATION OF SIZE
While factoring is primarily taught and practiced in algebra courses, it is used in every aspect of mathematics. Geometry is no exception. In the field of geometry, there exists the rule of similar triangles. The rule of similar triangles shows that if two triangles have the same angles and the lengths of two legs on one triangle along with a corresponding leg on the other triangle is known, there exists a common factor that can be used to determine the lengths of the other legs. For example, if one wishes to determine the height of a flagpole, factoring through the use of similar triangles can be employed. This is accomplished by an individual of known height standing next to the flagpole. The shadows of both the individual and the flagpole will now be measured. Because the person in standing perpendicular to the ground, a 90-degree triangle is formed with the height of the person being one leg, the length of the shadow being the other leg, and the hypotenuse being the distance from the tip of the person's head to the tip of the head on the shadow. The flagpole forms a similar 90-degree triangle. Once the lengths of the shadows are known, divide the length of the flagpole's shadow by the length of the individual's shadow to determine the common factor. This factor is then multiplied by the height of the individual to find the height of the flagpole.
Potential Applications
In engineering, business, research, and even entertainment, factoring can become a valuable asset. Engineers must use factoring on a daily basis. The job of an engineer is either to design new innovations or to troubleshoot problems as arise in existing systems. Either way, engineers look for effective solutions to complex problems. In order to make their job easier, it is important for them to be able to identify the problem, the solution, and—with regard to the mathematics that describe the systems and events—the factors that systems and events share. Once equations describing systems and events are factored, the most essential elements (the elements that unite and separate systems) can often be more clearly identified. The relationship of each component in the problem will often lead to the solution.
In business, factoring can help identify fundamental factors of cost or expense that impact profits. In research applications, mathematical factoring can reduce complex molecular configurations to more simplified representations that allow researchers to more easily manipulate and design new molecular configurations that result in drugs with greater efficiency—or that can be produced at a lower cost. Factoring even plays a role in entertainment and movie making as complex mathematical patterns related to movement can be factored into simpler forms that allow artists to produce high quality animations in a fraction of the time it would take to actually draw each frame. Factoring of data gained from sensors worn by actors (e.g., sensors on the leg, arms, and head, etc.) provide massive amounts of data. Factoring allows for the simplified and faster manipulation of such data and also allow for mapping to pixels (units of image data) that together form high quality animation or special effects sequences.
