Algebraic Expressions - Variables, Terms and Expressions

Algebraic Expressions - Variables, Terms and Expressions

Algebraic Expressions

What is an Algebraic Expression?

The combination of constants and variables, connected by signs of fundamental operations (+, -, ×, ÷) is called an algebraic expression.

For example, 2x – 3y + 9z is an algebraic expression.

In the algebraic expression 5x² + 7y³ - 4xy, 5x², 7y³, -4xy are called terms of the expression.

Variables, Terms and Expressions

In mathematics, often the value of a certain number may be unknown. A variable is a symbol, usually a letter, which is used to represent an unknown number.
Some examples of variables are: x, a, t, y, b

term can be a number, a variable, or a number and variable combined by multiplication or division.
Some examples of terms are: x, 8, 4y

An expression can be termed as a collection of terms separated by addition or subtraction operations. Some examples of expressions, with the numbers of terms, are listed below:

 Expression Number of Terms Description 3x 1 A number multiplied by a variable. The number is always written first followed by the variable(s). 2w – 8 2 Terms separated by – 5b + 7t – 6 3 Terms separated by + and – 5y/x 1 All multiplication and division, no + or – symbol

Classification of Algebraic Expressions

Monomial

An algebraic expression containing only one term is called a monomial.

For example, 2x, -5, ⁵/₉ abc are all monomials.

Binomial

An algebraic expression containing exactly two terms is called a binomial.

For example, 3x – 7, 4x + 9y, ab + c are all binomials.

Trinomial

An algebraic expression containing exactly three terms is called a trinomial.

For example, 3x – 2y + 7, 4x + 7y – 5z, a4 + b² + c3 are all trinomials.

Multinomial

An algebraic expression containing two or more terms is called a multinomial.

For example, 2x³ y² + 5x²y – 3xy + 7, 3a² + b² - 4c² - d², 2l + 3m + n – 5p are all multinomials.

Polynomial

In an algebraic expression, if the power of variables is a non-negative integer, then that expression is called a polynomial.

For example, 2x² + 5x + 7 is a polynomial.

2x² + 3/x is not a polynomial.
[The power of x in 3/x is negative. Therefore, 3/x = 3x−1]

3√x + 2x² - 5 is not a polynomial.

[The power of x in 3√x is in fraction. Therefore, 3√x = 3x1/2]

Degree of the Polynomial

In 2x² + 5x + 7, the degree of the polynomial is 2 as the highest power of the variable is 2.
Thus, the highest power of the variables in a polynomial is called the degree of the polynomial.

Example: Find the degree of the following polynomials.
a.  a5 + a3b3 + b4              b. x3y7 – x8 + y5            c. x9 + x4y3z7 – y8 + z7

Solution:
a.  a5 + a3b3 + b4
Degree of first term = 5
Degree of second term = 3 + 3 = 6
Degree of third term = 4
Thus, the degree of the polynomial is 6.

b. x3y7 – x8 + y5
Degree of first term = 3 + 7 = 10
Degree of second term = 8
Degree of third term = 5
Thus, the degree of the polynomial is 10.

c. x9 + x4y3z7 – y8 + z7
Degree of first term = 9
Degree of second term = 4 + 3 + 7 = 14
Degree of third term = 8
Degree of fourth term = 7
Thus, the degree of the polynomial is 14.

To add two or more algebraic expressions, we make groups of like terms and then find the sum of like terms in each group.

Example 1: Find the sum of 2a – 5b – 7c, 4a – 3b + c and a + 2b + 3c.
Solution: (2a – 5b – 7c) + (4a – 3b + c) + (a + 2b + 3c)
= (2a + 4a + a) + (– 5b – 3b + 2b) + (– 7c + c + 3c)
= (2 + 4 + 1)a + (– 5 – 3 + 2)b + (– 7 + 1 + 3)c
= 7a – 6b – 3c

Example 2: Add 5x + 3y – 4z, 12x + 4y + 3z and –7x – 5y + 2z.
Solution: (5x + 3y – 4z) + (12x + 4y + 3z) + (–7x – 5y + 2z)
= (5x + 12x – 7x) + (3y + 4y – 5y) + (–4z + 3z + 2z)
= (5 + 12 – 7)x + (3 + 4 – 5)y + (–4 + 3 + 2)z
= 10x + 2y + z

Subtraction of Algebraic Expressions

Change the signs of the terms of expression to be subtracted and add the two expressions so obtained by collecting different groups of like terms of both the expressions.

Example 1: Subtract (3a + 2b + 3c) from (5a – 3b + c).

Solution:
(5a – 3b + c) – (3a + 2b + 3c) = 5a – 3b + c – 3a 2b 3c
= (5 – 3)a + (– 3 – 2)b + (1 – 3)c
= 2a – 5b – 2c

Example 2: Subtract (5x – 5y + 7z) from (8x – 3y – 4z).

Solution:
(8x – 3y – 4z) – (5x – 5y + 7z) = 8x – 3y – 4z – 5x + 5y – 7z
= (8 – 5)x + (–3 + 5)y + (–4 –7)z
= 3x + 2y – 11z

Simplification of Algebraic Expressions

Expressions are put into their simplest form so as not to be confusing or too complex. One way of simplifying expressions is to combine like terms. By combining like terms, we can shorten and simplify our expressions, making them easier to read. Like terms often contain the same variable or variables.

Example 1: Simplify:
a. 8x – 8 + 3(2x – 5)           b. 7y2 – y – (3y2 – 6y) + 8y          c. 12x3 + 5 – [5(x3 – 3) + 12]

Solution:
a. 8x – 8 + 3(2x – 5) = 8x – 8 + 6x – 15
= 14x – 23

b. 7y2 – y – (3y2 – 6y) + 8y = 7y2 – y – 3y2 + 6y + 8y
= 4y2 + 13y

c. 12x3 + 5 – [5(x3 – 3) + 12] = 12x3 + 5 – [5x3 – 15 + 12]
= 12x3 + 5 – [5x3 – 3]
= 12x3 + 5 – 5x3 + 3 = 7x3 + 8

Example 2: Simplify: 13 + [a – {4b – (6a + b – 5) + 2a}] – {a – (b – 5)}

Solution: We have, 13 + [a – {4b – (6a + b – 5) + 2a}] – {a – (b – 5)}
= 13 + [a – {4b – 6a b + 5 + 2a}] – {a b + 5}
= 13 + [a – 4b + 6a + b – 5 – 2a] – a + b – 5
= 13 + a – 4b + 6a + b – 5 – 2a a + b – 5
= (1 + 6 – 2 – 1)a + (– 4 + 1 + 1) b + 13 – 5 – 5
= 4a – 2b + 3

Evaluation of Algebraic Expressions

A term is any coefficient and its variable multiplied together.
Identify the term in the expression: 6x – 2y + 5
The terms are 6x, 2y and 5.
Identify the coefficient and variable in the term: 4xy
The coefficient is 4 and variable is xy.

Example 1: If x = 3 and y = 2, evaluate 4x + 5y.

Solution: 4x + 5y = 4 × 3 + 5 × 2 = 12 + 10 = 22

Example 2: Evaluate the expression
(5 + a) – b, if a = 6 and b = 4.

Solution: (5 + a) – 3b = 5 + 6 – 4 = 7

Example 3: If x = 2, y = –1 and z = 3, find the values of the following expressions.
a. 5x2 + y2 + z2                  b. 2x + y + z

Solution:
a. Substituting x = 2, y = –1 and z = 3 in the expression, we get
5x2 + y2 + z2 = 5(2)2 + (–1)2 + (3)2
= 5 × 4 + 1 + 9 = 30
b. 2x + y + z = 2 × 2 – 1 + 3 = 6