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.
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
A 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.
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.
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.
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.
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.
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.
Addition of Algebraic Expressions
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]
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.
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