Algebraic Expressions - Formulas, Simplifying, Evaluating

What is an Algebraic Expression?

Algebraic expression is an expression that is made up of combining variables and constants, along with basic algebraic elementary operations like addition, subtraction, multiplication or division. It is the terms that finally make up an algebraic expression. So we can say that an expression is made up of various parts combined together.

For Example
$10 x+4 y-100,56 x-10$, etc.
We must note that unlike the algebraic equation, an algebraic expression has no sides or is equal to a sign. Some examples listed below:
- $90 \mathrm{x}+2 \mathrm{y}-50$
- $x-45$

In the above expression (i.e. $20 \mathrm{x}-70$ ),
- x is a variable, whose value is unknown to us and takes any random value.
- 20 is known as the coefficient of $x$, as it is a constant value used with the variable term.
- 70 is the constant value term that has a definite value.

Types of Algebraic Expression

There are 3 main types of algebraic expressions which include:

• Monomial Expression
• Binomial Expression
• Polynomial Expression

Monomial Expression: An algebraic expression that has only one term.

For example, $30 x^4, 6 x y$, etc.

Binomial Expression: An algebraic expression that has two terms, which are unlike.

For example, $5 a b+8, p q r+x^3$, etc.

Polynomial Expression: An algebraic expression with more than one term with non-negative integral exponents of a variable.

For example $a x+b y+c a, x^3+56 x+10$, etc.

An algebraic expression can also be categorised into two additional types as:

• Numeric Expression
• Variable Expression

Numeric Expression: It consists of numbers and operations, but do not include any variable. Few examples are $10+$ $67,15 \div 9$, etc.

Variable Expression: It contains variables along with numbers and operation to define an expression. For example $19 x+y, 23 a b+33$, etc.

We will come across the terms of algebraic expressions such as:

• Coefficient of a term
• Variables
• Constant
• Factors of a term
• Terms of equations
• Like and Unlike terms

If $20 x^2+30 x y+40 x+7$ is an algebraic expression.
Then, $20 x^2, 30 x y, 40 x$ and 7 are the terms
Coefficient of term $x^2=20$
Coefficient of term $x=40$
Coefficient of term $\mathrm{xy}=30$
Constant term $=7$

Now we define certain kinds of terms with their examples:

Like terms can be defined as those that have same variable. For example, $200x$ and $30x$
Unlike terms can be defined as those that have different variable. For example, $x$ and $35 y$
Factors of a term If 3 pq is a term, then its factors are $3, p$ and $q$.

Operations on Algebraic Expressions

The operations on algebraic expressions include operations like addition, subtraction, multiplication and division of algebraic expressions.

Addition and Subtraction of Algebraic Expressions

Any two or more algebraic expressions can be added and subtracted. We can add and subtract like terms of an algebraic expressions easily.

Example: Add $30 x+15 y-6 z$ and $x-40 y+2 z$.

By adding both the expressions we get;

$
(30 x+15 y-6 z)+(x-40 y+2 z)
$

Separating the like terms and adding them together:

$
\begin{aligned}
& (30 x+x)+(15 y-40 y)+(-6 z+2 z) \\
& 31 x-25 y-4 z
\end{aligned}
$

Multiplication of Algebraic Expressions

In this process, we multiply every term of the first expression with every term of the second expression and at last combine all the products. So, we go to one particular term in an expression and then perform the desired arithmetic operation with every other term of another expression. For example,

- $a b(4 a b+13)=4 a^2 b^2+13 a b$ Here ab is taken separately and is multiplied with each term of another expression which are $4ab$ and $13$.

Another example:
- $(y+1)(y+2)=y^2+y+2 y+2=y^2+3 y+2$

Division of Algebraic Expressions

We factor out the numerator and the denominator, cancel the possible terms, and simplify the rest. For example,

- $\frac{2 x^2}{\left(2 x^2+4 x\right)}=\frac{\left(2 x^2\right)}{[2 x(x+2)]}=\frac{x}{(x+2)}$ Here the first term is divided separately by each term in second expression and then final result is written.

Another example:
- ${\left(x^2+5 x+4\right)}{(x+1)}=\frac{[(x+4)(x+1)]}{(x+1)}=x+4$

Algebraic Expressions Formulas

Now, let us look into some algebraic expressions and identities.

Algebraic expressions and identities class 8

The general algebraic formulas used to solve the expressions or equations are:

- $(a+b)^2=a^2+2 a b+b^2$
- $(a-b)^2=a^2-2 a b+b^2$
- $a^2-b^2=(a-b)(a+b)$
- $(a+b)^3=a^3+b^3+3 a b(a+b)$
- $(a-b)^3=a^3-b^3-3 a b(a-b)$
- $a^3-b^3=(a-b)\left(a^2+a b+b^2\right)$
- $a^3+b^3=(a+b)\left(a^2-a b+b^2\right)$

Algebraic Expressions Class 8 Extra Questions

Now let us look into some algebraic expressions examples.

Example 1: There are 20 apples in a bag. Write the algebraic expression for the number of apples in $p$ number of bags.

Solution: The number of apples in one bag $=20$. The number of bags $=\mathrm{y}$. So the number of apples in y bags $=20 \mathrm{y}$.

Example 2: What type of algebraic expression is $40 \mathrm{x}+52$ ?
Solution: $40 \mathrm{x}+52$ has two monomials 40 x and 52 hence it is a binomial. Every binomial is a polynomial as well. So $40 x+52$ is a polynomial as well. So the correct answers are: binomial and polynomial.

Example 3: Is 22a/x a monomial expression? Justify your answer.
Solution: The expression has a single non-zero term, but the denominator of the expression is a variable. So it is not a monomial.

Example 4: Add the following algebraic expressions: $33 x+2$ and $44 y+2 z$.
Solution: The given algebraic expressions have no like terms. Hence their sum is $33 \mathrm{x}+2+$ $44 y+2 z$. If we rearrange the terms, we get the sum $=33 x+44 y+2 z+2$.

Example 5: Simplify the given algebraic expressions by combining the like terms and write the type of Algebraic expression.
(i) $30 x y^3+19 x^2 y^3+55 y^3 x$
(ii) $71 a b^2 c^2+21 a^3 b^2-31 a b c-53 a b^2 c^2-2 b^2 a^3+22 a b$

Solution: Creating a table to find the solution:

List of Topics Related to Algebraic Expressions