Home » Mastering Algebra - Recognizing Special Products and Their Factors - Part I

Mastering Algebra - Recognizing Special Products and Their Factors - Part I

Posted: Monday, May 02, 2011

by Joe Pagano
Math by Joe

One of the key difficulties in algebra is learning to recognize special products and their respective factors. For example, the special polynomials which are formed by the difference of two perfect squares and perfect square trinomials have standard factors which can easily be obtained once one recognizes these polynomials for what they are. Knowing the vocabulary of algebra goes a long way in mastering this often times difficult-for-high-school-kids subject. Let's look at some of these special products and master them so they never bother you again.

Algebra consists of essentially two things: constants and variables. A constant is nothing more than a number whose value is, well constant; the value never changes. A variable, on the other hand, stands for a constant; its value varies. These constants and variables can be joined in a dizzying array of ways to form all kinds of simple and, yes, complicated expressions. To categorize the vast number of expressions, mathematicians give names to certain expressions which meet certain criteria. Let us explore some of these.

To wit, a polynomial-an oft encountered expression in algebra-is a combination of constants and the same variable so that the expression looks this: a(n)x^(n) + a(n-1)x^(n-1) + a(n-2)x^(n-2) + ... + a(1)x + a(0), where each of the a(n)'s stands for some number like 3, 4, or -2, and the "^" symbol represents "to the power of." That is if n is 5, then we are raising the variable x to the 5th power, then the 4th, and descending by 1 each time until we get to the 1st and then 0th power. (Remember any expression or number to the 0th power is always 1). Examples of polynomials are any constant like 2 or 3; any variable to any power, like x^2 or 5x^5; and all combinations of constants and a given variable to different integral powers. (Remember: in order to classify as a polynomial we only allow positive integer exponents; that is fractions and negative integers are not allowed as exponents.)

Polynomial comes from the Greek word "poly" which means many. Thus a polynomial has many terms. We also have polynomials which are binomials-two terms, from "bi" = two; trinomials, which have three terms-from "tri" = three. Examples of binomials are (x - 4), (x + 5), and (3x - 4). Examples of trinomials are 2x^2 - 3x + 4and x^2 + 2x + 1. (Remember that 2x^2 means 2 times x times x again.)

Once we understand the aforementioned terminology, we can progress into mastering special products and factors, and also see how these come into play in applications in the real world. Then we will be in a position to recognize these special products and instantly be able to decompose them into their component factors. Stay tuned...

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