In mathematics, an algebraic expression is an expression built up from constants, variables, and a finite number of algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, is an algebraic expression. Since taking the square root is the same as raising to the power ,
is also an algebraic expression.
A rational expression is an expression that may be rewritten to a rational fraction by using the properties of the arithmetics operations (commutative properties and associative properties of addition and multiplication, distributive property and rules for the operations on the fractions). In other words, a rational expression is an expression which may be constructed from the variables and the constants by using only the four operations of the arithmetic. Thus, is a rational expression, whereas is not. A rational equation is an equation in which two rational fractions (or rational expressions) of the form are set equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected. Algebra has its own terminology to describe parts of an expression:
1 – Exponent (power), 2 – Coefficient, 3 – term, 4 – operator, 5 – constant, - variables Variables
By convention, letters at the beginning of the alphabet (e.g. ) are typically used to represent constants, and those toward the end of the alphabet (e.g. and ) are used to represent variables. They are usually written in italics. Exponents
By convention, terms with the highest power (exponent), are written on the left, for example, is written to the left of . When a coefficient is one, it is usually omitted (e.g. is written ). Likewise when the exponent (power) is one, (e.g. is written ), and, when the exponent is zero, the result is always 1 (e.g. is always )
A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as x2 + 4x + 4. An irrational algebraic expression is one that is not rational, such as √x + 4. 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 as place 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 a variable 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 calledTerms 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, 5x2 , 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)
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