Study Notes on Algebraic Expressions
Sean Anthony Kyle Antonio
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Algebraic expression

History:
Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by medieval times Islamic mathematicians were able to talk about arbitrarily high powers of the unknown x, and work out the basic algebra of polynomials (without yet using modern symbolism). This included the ability to multiply, divide, and find square roots of polynomials as well as knowledge of the binomial theorem. The Persian mathematician, astronomer, and poet Omar Khayyam showed how to express roots of cubic equations by line segments obtained by intersecting conic sections, but he could not find a formula for the roots. A Latin translation of AlKhwarizmi's Algebra appeared in the 12th century. In the early 13th century, the great Italian mathematician Leonardo Fibonacci achieved a close approximation to the solution of the cubic equation x3+ 2x2 + cx = d. Because Fibonacci had travelled in Islamic lands, he probably used an Arabic method of successive approximations.

Examples :
Problem:  Ms. Jensen likes to divide her class into groups of 2. Use mathematical symbols to represent all the students in her class.  Solution:  Let g represent the number of groups in Ms. Jensen's class.   Then 2 · g, or 2g can represent "g groups of 2 students". 
In the problem above, the variable g represents the number of groups in Ms. Jensen's class. A variable is a symbol used to represent a number in an expression or an equation. The value of this number can vary (change). Let's look at an example in which we use a variable.
Example 1:  Write each phrase as a mathematical expression. 
 Phrase Expression
the sum of nine and eight 9 + 8
the sum of nine and a number x 9 + x

The expression 9 + 8 represents a single number (17). This expression is a numerical expression, (also called an arithmetic expression). The expression 9 + xrepresents a value that can change. If x is 2, then the expression 9 + x has a value of 11. If x is 6, then the expression has a value of 15. So 9 + x is analgebraic expression.
Example 2:  Write each phrase as an algebraic expression.  
 Phrase Expression
nine increased by a number x 9 + x
fourteen decreased by a number p 14  p
seven less than a number t t  7
the product of 9 and a number n 9 · n or 9n
thirtytwo divided by a number y 32 ÷ y or 
 
In Example 2, each algebraic expression consisted of one number, one operation and one variable. Let's look at an example in which the expression consists of more than one number and/or operation.
Example 3:  Write each phrase as an algebraic expression using the variable n.  
 Phrase Expression
five more than twice a number 2n + 5
the product of a number and 6 6n
seven divided by twice a number 7 ÷ 2n or 
three times a number decreased by 11 3n  11
 
Example 4:  A small company has $1000 to distribute to its employees as a bonus. How much money will each employee get?  Solution:  Let e represent the number of employees in the company. The amount of money each employee will get is represented by the following algebraic expression:   
  
Example 5:  An electrician charges $45 per hour and spends $20 a day on gasoline. Write an algebraic expression to represent his earnings for one day.  Solution:  Let x represent the number of hours the electrician works in one day. The electrician's earnings can be represented by the following algebraic expression:   
Solution:  45x  20 
Summary:  A variable is a symbol used to represent a number in an expression...
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