Polynomials, Mathematics

In arithmetic, we deal with numbers. In contrast to this, in algebra, we deal with symbols. These symbols are often represented by lower case alphabets. One of the advantages of using alphabets is that they are easily identifiable and make some sense even to a lay person. As against figures which stand for a definite value under all the circumstances, symbols in algebra are situation specific. That is, x in a certain problem may take 2 as its value while in some other problem it may take 3 or 11 or 154 as its value, while numbers remain the same whenever and wherever they are used. Another important facet of algebra is that we can operate with symbols themselves without assigning them any value whatsoever. The basic signs (+, -, x,  / ) or operators as they are otherwise known as, have the same meaning they hold in arithmetic. In algebra, we can have terms like 'a' or '7q' or '7p + b' or 'a + 8b - c'. If we consider terms like a + 8b - c or 7p + b, we find that they are a collection of symbols separated by signs and they may contain either one or two or more such symbols. All such collection of terms separated by signs are generally referred to as algebraic expressions. And as you can see expressions may be simple or compound. A compound expression can be a binomial, trinomial or a polynomial expression. An expression like 'a' or '7q' is referred to as simple expression whereas expressions like 7p + b is referred to as binomial expression as it contains two terms and the one similar to a + 8b - c is referred to as a trinomial (it is a collection of three terms). Also expressions containing more than three terms are referred to as multinomials or polynomials.

When two or more quantities are multiplied with each other the resultant number is referred to as the product of those quantities. The point to observe in this case is that while in arithmetic the product of two numbers is shown as 2 x 3, the same in algebra can be shown as ab or a x b or a.b. All these notations convey the same meaning. Now consider a quantity 8abc. This is a result of multiplying 8, a, b and c. These quantities are referred to as the factors of the product 8abc. Also in this product, the number 8 is referred to as a coefficient of the remaining factors. In the broad sense of it even the quantity 'ab' is also referred to as a coefficient. But to differentiate it from a numerical coefficient, we refer to it as a literal coefficient. Also the product 8abc can be expressed as 8bac or 8cab. Only one prefers to present them in alphabetical order.

Now how do we express a quantity like a.a.a.a. We observe that 'a' has been multiplied by itself four times. The product of a quantity when multiplied by itself repeatedly is usually referred to as the power of that quantity and is expressed by writing the number of factors to the right of that quantity and above it. It will be like a4. When a quantity is expressed in this form, the quantity 'a' is referred to as base and the numerical 4 is referred to as the power or index or exponent. As we treat a number without a sign as positive, any number whose power is not specified is understood to have its power as 1. Here one should understand the difference between the coefficient and power very clearly.

Usually the quantities like a2 is read as 'a squared', and a3 as 'a cubed' and so on. With this background we should be able to solve problems like finding the value of 5abc + 7d given that a = 2, b = 1, c = 3 and d = 3. It will be  5.2.1.3 + 7.3 That is,     30 + 21 = 51

Posted Date: 9/13/2012 2:31:00 AM | Location : United States







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