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 Al-Khwarizmi’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+ 2×2 + 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. | | |
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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| thirty-two 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| he 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 or an equation. The value of this number can change. An algebraic expression is a mathematical expression that consists of variables, numbers and operations. The value of this expression can change. | ————————————————- Algebraic Expression Definitions 1. Algebraic Expression:
An expression consisting of arithmetic numbers, letters (used as symbols) and operation signs is called an Algebraic Expression Examples: 2x + 3y , -9p + 2r, x2 + 5x + 6, a3 + b3 + 3ab2 + 3a2b 2. Constant: Algebraic symbols that have a fixed value and do not change like variables (which are used asplace holders) are called Constants Examples: In 2x + 3y + 4, 4 is a constant. In 2a2 – 3ab + 7, 7 is a constant 3. Variable A symbol in Algebra that can be plugged in with different numerical values (numbers) is called avariable In 5p + 6q + r, the letters (symbols) p,q are called Variables.
Note: 5p + 6q + r is also a variable, since any number can be plugged in for p, q and r as required. 4 . Terms of an expression The parts in an algebraic expression connected by the operation signs + or — are called Terms In 2y + 3, 2y is one term and 3 is another term. 5. Monomials An algebraic expression containing only one term is called a Monomial. Monomials are also called simple expressions. 2x, 5×2 , pq are examples of monomials. 6. Binomial An algebraic expression that contains two terms is called a Binomial 2x + 3y, 2p2 + 9y3 are some examples of Binomials. 7. Trinomial
An algebraic expression that has three terms is called a Trinomial. 3x + 4y + 5z, ax2 + bx + c are examples of Trinomials. 8. Polynomial An algebraic expression that contains one term, two terms, three terms or more is called aPolynomial. By this definition, each of monomial, binomial and trinomial is a Polynomial. Examples of Polynomials are: 3x, 4y2, pq, (which are Monomials) 3x + 4y, 5q + 9t, -m2 – n2 (which are Binomials) ax2 + bx + c, 3a – b + (5/3) c. (which are Trinomials) 9. Factor: Symbols or Numbers in multiplication are called factors. Example: In pq, p and q are factors in the multiplication p ? . pq is called the product of the factors p and q. Example: 2 and 3 are factors in the multiplication 2? 3 which is equal to 6. 2 and 3 are called Numerical Factors 6 is called the product of 2 and 3. In the product xy, x and y are the factors and xy is called the product of x and y. x and y are called Literal Factors. 10. Coefficient: Coefficient is of two types. Numerical coefficient and Literal coefficient. Numbers form Numerical coefficients and symbols form literal coefficients. Examples: In 2xy, 2 is the number or the Numerical coefficient while xy, the symbol, is the Literal Coefficient.
In the Monomial y, the Numerical coefficient is 1 and the literal coefficient is y In the product 100xy, 100 is the Numerical coefficient and xy is the literal coefficient. 11. Like Terms: Two or more terms that have the same literal coefficients are called Like Terms. Like terms can have different Numerical Coefficients, but not literal coefficients. Examples: 4pq and 100pq are like terms as the literal coefficients xy are same in the two terms. -13p2q2 and 13 p2q2 are Like terms as only the numerical coefficients are different but the literal coefficients are same. 12. Unlike Terms
Terms that are not Like terms are Unlike Terms. So, Unlike terms have different literal coefficients. Examples 3xy, 3xy2 are unlike terms . Algebraic Expression: Any expression that contains letters, numbers and basic operation signs +, –, ? and ? is called an Algebraic Expression. Examples: 2x, 5y, -20p, 3x + 5y, -9q/8 are a few examples of algebraic expressions. In the above examples, x, y, p, q are the letters and 2, 5, -20, -9/8 are the numbers, while the symbols: -, +, ? are the basic (fundamental) signs of operations. The Letters are called variables and the numbers before them are called coefficients. . Letters used as symbols for numbers: In the algebraic expression 2x, the letter x stands as a symbol for any number. One can choose any number to write for x . So, x holds a place for any number. One can write 3, 10, 100 or any other number as required for x . Since x changes or varies based on what is to be written for it, it is therefore calledVariable. What varies is a variable. Other variables in the above examples for algebraic expression are p, q and y 2. Symbols used to denote Multiplication 1. Now consider 2x. What does it mean? It means multiplication of 2 and x, i. . , 2 ? x. 2 ? x is also called product of 2 and x. Product refers to multiplication. In 2x we know x is the variable and we also know it stands as a symbol to write any number for it. In other words, it holds a place to write any number for it. Then let us see what will 2x become, when numbers like 3, 10, 100 are written for x 2x means product (multiplication) of 2 and x. So, we get 2 ? 3 = 6, 2 ? 10 =20, 2? 100 =200 2. Again consider 2x It may also be written as 2. x standing for multiplication (product) of 2 and x. so, we have 2. x = 2. 3 = 6 or 2. 10 = 20 or 2. 00 = 200, depending on what number is written for x. In algebraic expressions, dot indicates multiplication 3. Consider 2x once again. It can also be written as 2(x) standing for multiplication of 2 and x So, we have 2(3) = 6 or 2(10) = 20 or 2(100) =200, based on what number we choose to write for x Let us summarize the above three ways of representing multiplication in the table below: Symbols used to denote Multiplication in Algebraic Expression 1. A dot placed between symbols used for numbers 2. A parentheses between symbols or numbers 3. Writing no operation sign between symbols (not numbers)
Consider examples to understand the three forms of Algebric Expression: 1. A dot placed between symbols 2. x = 2x, x. y = xy, 7. p = 7p and other examples 2. A parentheses between symbols or numbers 2(x) = 2x, x(y) = xy, 7(p) = 7p 3. Writing no operation sign between symbols (not numbers) 2x means 2? y xy means x? y 7p means 7? p Important Note: Number 50 does not stand for product of 5 and 0 i. e. , it is not 0, it is just the number 50. Only in Algebraic Expressions symbols like x and y not connected by any operation sign stand for multiplication of x and y. ————————————————————————————————————— 3. Symbols used to denote Division To denote division of x and y, we write x? y. The algebraic expression x / y also stands for division of x by y 4. Converting Words into Symbols 5. Use of Parentheses 1. To stand for multiplication: 2(x) means 2 ? x, i. e. the product of 2 and x 2(3) = 2 ? 3 = 6 x (y) = x ? y 2. To consider an Algebraic Expression as one number 3(p + q) is one number i. e. , the product of 3 and sum of x and y 6. The Substitution Method:
It’s a very important method. It’s the basic method to find values of variables which stand as place holders. Let us see it below: Find the value of each of the following algebraic expressions if x = 2 and y = 3 1. x + y Solution: just plug in 2 for x and 3 for y and called this substitution. x + y = 2 + 3 = 5 2. 3x + 4y Solution: as in 1. above, just plug in 2 for x and 3 for y 3. 2 + 4. 3 = 6 + 12 = 18 (recall that 3x stands for multiplication of 3 and x and so also 4y) 3. (2x – 3y)/(3x + 4y) Solution: again as in 1 and 2 above, write the given values in x and y