Algebraic Expressios

Algebraic expressions are formed by terms which contain both Variables and constants.

                         4x + 5, 8p – 9, 10ab – 2c and 6x² + 7 

are examples of algebraic expressions.

In the algebraic expression 4x + 5, 4x and 5 are terms of the algebraic expression. A term may be a constant or a product of two or more constants and variables. For instant, the term 4x is the product of two factors, 4 and x, while the term 5 is a constant.

The numerical factor of a term is called its coefficient. The coefficient of x in 4x + 5 is 4. In expression 10ab – 2c, the coefficient of ab is 10 and coefficient of c is – 2.

What is coefficient if m in m + 2n + 4? Is it zero or 1? It is 1.

The value of the variable p in 7p + 3 can be anything. It can be 1, 2, 3, 4, 0, - 2 , - 3 etc. The value of the expression also changes depending on the value of p. If p = 1, the value of the expression is 10; if p = 6, the value of an expression changes with the value chosen for its variable.

ALGEBRAIC EXPRESSIONS ON THE NUMBER LINE

An algebraic expression does not have any specific position on the number line. However, if the variable of the expression is assigned a place on the number line, the position of the expression relative to the variable is well defined on the number line.

For instance; if the variable y of the expression y + 8 is represented by the point. Y on the number line as shown, then the expression y + 8 is at a distance of 8 units to the right of Y. Similarly, y – 3 at a distance of 4 units to the left of Y.

                                                     Y

  <-------------------------------|---|---|---|---|---|---|---|---|---|---|---|----------------------------->

                                   Y – 3          y                                + 8

We can also represent expressions, like 3y + 3 on the number line if we know the position of the variable y on the number line.

Assume variable y is represented by Y, then 3y is represented by P, which is at a distance equal to four times the distance of Y from O. The expression 3y + 3 is represented by Q which is at a distance of 3 units to the right of 3y.

                                           O   Y                  P          Q

          <---------------------------|---|---|---|---|---|---|---|---|---|--------------------------->-

                                                                    3y         3y + 3-  

Like and unlike Terms

Terms of an algebraic expression are said to be like terms if they have the same variable. The variable must also have the same power in the said terms. However, the coefficients of the variable may be different. Terms either not having the same variable or those with same variable but unequal powers are called unlike terms

Like Terms                                                           Unlike Terms

5a, 16a, - 6a                                                        12a, 10b, 23c,

2xy, 4xy, 8xy                                                       6x³, x², x 6x⁰

Monomials, Binomials and Polynomials

An algebraic expression is said to be a monomial if it contains only one term. The following expressions are all monomials.

                                                      4xy, 11abc, 7 axby, - 6xyz, 2a² b²

An algebraic expression is said to be a binomial if it contains two unlike terms. e.g.,

                                                       ax +b , 10a + 6b, 15q² + 6

An algebraic expression is said to be a trinomial if it contains three unlike terms. eg.,

                                                      2x + 3y +4z, 2m² + 5n² + o²

In general, an algebraic expression, which contains one or more terms with nonzero coefficients and variables having nonnegative integral exponents, i.e., power of the variable, is called a polynomial. e.g.,

                                                     2x + 3y +4z + 9, 5abc, 4x² + 6y² +13