Algebraic Expression

Algebraic expression

An algebraic expression is a mathematical statement or phrase formed by combining numbers, variables, constants, and mathematical operations.

Constant

A constant is a symbol that represents a fixed value, it is generally denoted by \(a, b, c, \dots\).
For example, in the equation \(y = ax + b\), the symbols \(a\) and \(b\) are constants.
Some more example of constant are

1. The number of provinces in Nepal.
2. The number of the capital city of Nepal.

Variable

What is a Variable?

A variable is a symbol that represents a value which can change or vary. It is commonly denoted by letters such as \(x, y, z, \dots\).
For example, in the equation \(y = ax + b\), the symbol \(x\) is a variable because its value can change.
Some more examples of variables are

1. The name of provinces in Nepal.
2. The number of districts in provinces in Nepal.

S. No.	Expression	Variable Term	Constant term
(i)	\( xy + 4 \)	\( xy \)	\( 4 \)
(ii)	\( 13 - y^2 \)	\( -y^2 \)	\(13 \)
iii)	\( 13 - y + 5y^2 \)	\(- y + 5y^2 \)	\(13 \)
(iv)	\( 4p^2q - 3pq^2 + 5 \)	\( 4p^2q - 3pq^2 \)	\(5 \)

Term

A term is any single mathematical symbol, which can include a number, a variable, or a constant. For example: \(5, x, 3y, 7x^2\). Terms can be classified as follows

1. Constant term: A term with a fixed value and no variable (e.g., \(5\)).
2. Variable term: A term that includes a variable. It may have either a constant or a variable as its coefficient (e.g., \(3y\) or \(y\)).

S. No.	Expression	Number of Term(s)
(i)	\( xy + 4 \)	2
(ii)	\( 13 - y^2 \)	2
iii)	\( 13 - y + 5y^2 \)	3
(iv)	\( 4p^2q - 3pq^2 + 5 \)	3

Algebraic expression

An algebraic expression is a combination of one or more terms joined by operations such as addition, subtraction, multiplication, or division, and wwhich contains variable.
For example
\(3x + 4\), \(2y^2 - y + 7m\), \(5a \times 3b\)
are algebraic expressions.

Is 4 an algebraic expression?

No, 4 is not an algebraic expression because an expression should have at least one variable. So, variables are used to study algebra, thereby distinguishing the topic from the study of arithmetic. For example

• \(2 \times (5+7)=2 \times 5 + 2 \times 7 \)
  is arithmetic statement about a specific set of numbers
• \(x \cdot (y+z) = x\cdot y + x\cdot z\)
  is an algebraic statement about any three set of numbers

Example

Bidhan start with Rs 1000 and save Rs 100 each week, what can be its algebraic expression to model the total amount Bidhan save?
Let \(w=\)number of weeks

Relate	Starting amount	plus	amount saved	times	number of weeks
Write	1000	+	100	\(\times\)	w

Therefore, the expression \(1000+100w\) models the situation.

Expressions are mathematical language, which is used to write word problems in mathematical terms. For example

In words	Expression
A number \( x \) is increased by 7	\( x + 7 \)
A number \( y \) is decreased by 7	\( y - 7 \)
A number \( a \) is multiplied by 7	\( a \times 7 \)
A number \( k \) is divided by 7	\( k \div 7 \)

Base and Coefficient

The base of an algebraic term is the main quantity or variable in the term, for example, in the term \(3x^2\), the symbol \(x\) is called base, 3 is called cofficient.

1. A coefficient is the numerical factor of a term that contains a variable as its base. It indicates how many times the variable is multiplied.
   For example, in the term \(3x^2\), the coefficient is \(3\), which means that \(x^2\) is multiplied by 3.
2. In terms, any multiplier can be considered the coefficient of the remaining factors. For example, in \(6abc\), the number \(6\) is the coefficient of \(abc\), and \(6ab\) is the coefficient of \(c\). Similarly, other combinations are also possible.

1. Based on the presence of variables or constants, algebraic expression can be either variable or constant.
2. Based on the number of terms, algebraic expression can be classified as a monomial, binomial, trinomial, or polynomial. According to the degree (exponent) of the expression, it can be linear, quadratic, cubic, and so on.

Like and Unlike Terms

In an expression, if the terms have the same literal (alphabetical) factors or the same variables with the same powers, they are called like terms.
For example,
\(4x^2\) and \(5x^2\) are like terms.
Unlike terms are those that do not have the same literal factors or variables with the same powers. For example,
\(5m\) and \(2n\) are unlike terms.

📝 Select "Same" or "Different" for variable Parts (literal factors), and "Like" or "Unlike" for Term Classification.

SN	Pair	Factors	Variable Parts	Term Classification
(i)	7x	7, x	--Select-- Same Different	--Select-- Like Unlike
	12y	12, y
(ii)	3xy	3, x, y	--Select-- Same Different	--Select-- Like Unlike
	3x	3, x
(iii)	15x	15, x	--Select-- Same Different	--Select-- Like Unlike
	-21x	-21, x
(iv)	6x2y	6, x, x, y	--Select-- Same Different	--Select-- Like Unlike
	9xy2	9, x, y, y