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5b1. Algebraic Expressions and Operations
A variable is a symbol that denotes a numerical quantity. Variables are alphabetic characters. Algebra utilizes the concept of a variable to generalize mathematical concepts to a particular group of numbers.


The number that the variable represents is called its value.


A constant is a symbol that represents a definite mathematical quantity.


e.g., 3, 4.9, 2/3, etc. Symbols such as P and e are also constants, approximately denoting the values 3.14 and 2.718 respectively.


A term is a product with an unspecified number of factors, where the factors are variables or constants. The variables of a term are called literal factors, and the constants the numerical coefficient (or simply coefficient), of the term. Terms that have the same literal factors which differ only in their numerical coefficients are called similar terms.


e.g., 3xy and 2xy are similar terms


An algebraic expression is a mathematical statement that may use constants and variables. E.g. 5x + 2 Denny weighs 70 kilograms, and Diana weighs k kilograms. The combined weight in kilograms of these two people is the sum of their weights, which is 70 + k.


By applying the distributive property, two or more similar terms can be combined into one term. The new term has the same literal factors as the similar terms, but its coefficient is the sum of the coefficients of the similar terms. This process is known as combining similar terms. 3xy + 2xy = 5xy


A monomial is a term containing variables with only nonnegative integers as exponents.
e.g., 4x, 5x2y.


An expression with two terms is a binomial; with three terms is a trinomial. A polynomial has more than one term.


The degree of the polynomial is the highest power of the variable in the terms.


The general form of a polynomial p(x) is:



Example


1. If (x + 1)2 = 100, which of the following could be the value of x - 4?


(A) -5   (B) 4   (C)-15   (D) 15  (E) None of these 


Solution: (x + 1)2 = 100 Þ x + 1 = ± 10


(i) x + 1 = 10 Þ  x = 9 and x - 4 = 5


(ii) x + 1 = -10 Þ  x = -11 and x - 4 = -15


Hence, (C) is the best choice.








 
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