Mathematics consists of a lot of concepts expressed in terms

Mathematics consists of a lot of concepts expressed in terms of symbols, formulas and other mathematical notation. For example, the exponent rule xmxn = xm+n. But what is this really saying?When multiplying the same base you add the exponents. The point is you should try to verbalize the meaning. This week you will have covered a variety of concepts such as polynomials, exponents, and factoring.Pick one of the topics you have covered and share with us your interpretation of some of definitions, rules, and methods used in solving. Also work out a non-trivial problem and try and explain each step of the process in your own words.

Solution

Polynomials are sums of and expressions of variables and exponents. Polynomial terms have variables which are raised to whole-number exponents or, the terms are just plain numbers. Any term that doesn\'t have a variable in it is called a \"constant\" term because, no matter what value you may put in for the variable x, that constant term will never change. The first term in the polynomial, when it is written in decreasing order, is also the term with the biggest exponent, and is called the \"leading term\". The exponent of a term tells us the \"degree\" of the term. The degree of the leading term tells us the degree of the whole polynomial. For example, x² - 6x + 8 is a polynomial of the second degree. There are the following 2 ways of solving an equation of 2^nd degree (also called a quadratic equation):

1^st Method:

We factorize the polynomial expression. For, example, if x² - 6x + 8 = 0, then x² - 2x – 4x + 8 = 0 or, x (x - 2) - 4 (x - 2) = 0 or, (x – 2) (x – 4) = 0 . Therefore, either x – 2 = 0 or, x – 4 = 0 so that either x = 2 or x = 4.

2^nd Method:

We use a formula for solving a quadratic equation. The solution of the quadratic equation ax² + bx + c = 0 is x = [ -b ± ( b² – 4ac)]/ 2a. Here a = 1, b = -6 and c = 8. Therefore, we have x = [ - (-6) ± { ( -6)² – 4(1)(8)}]/ 2*1 or, x = [ 6 ± ( 36 -32)]/ 2 or, x = (6 ± 4)/ 2 or, x = (6 ± 2)/2 . Thus, either x = (6 +2)/2 = 8/2 = 4 or, x = ( 6 – 2)/2 = 4/2 = 2.