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# Algebraic Expression Formula

Algebra is one of the most important sub-topics in mathematics. It has many academic and practical applications. In this topic, we will discuss such concepts with algebraic expression formula and some examples. Let us learn algebra!

## Algebraic Expression Formula

### What are Algebraic Expressions?

Numeric expressions apply operations to numbers. For example, 2 × (3 + 8) is a numeric expression.  But, if expressions include the variables and algebraic terms then it will become algebraic expression. Thus algebraic expressions include `with one variable and with minimum one operation like addition, subtraction, multiplication, division, etc.  For example, 2 × (x + 8y) is an algebraic expression.

An algebraic expression is an expression that consists of constants, variables, and some algebraic operations. For example, $$3x^2 − 2xy + d$$  is an algebraic expression. Thus, an algebraic expression is built up from three types of basic elements:

• Coefficient (i.e. numbers)
• Constants (letters at the beginning of the alphabet, such as  a, b, c )
• Variables (letters at the end of the alphabet, such as  x, y, z )

As we can see that the constants and variables are letters. But they are representing as numerical values. There is an important difference between the constants and variables. A constant is a value that remains fixed. But it is unknown or undetermined. A variable is a value that may change.

As we can see above, except for the three basic elements, an algebraic expression is also built up from algebraic operations. These operations are addition, subtraction, multiplication, division, exponentiation, etc.

### Types of Algebraic Expression

There are 3 mains types of algebraic expressions which include:

• Monomial Expression: Such algebraic expressions are having only one term. Examples of monomial expressions include $$3x^4 , 3xy, 4x^3 + 3y^4$$
• Binomial Expression: A binomial expression is an algebraic expression which is having two different terms. Examples of binomial include 5xy + 8, xyz + x3, etc.
• Polynomial Expression: In general, an expression with more than one term some integral exponents of a variable is called a polynomial.
• Numeric Expression: A numeric expression consists of numbers and operations, but never include any variable. Some of the examples of numeric expressions are 10+5, 15 × 2, etc.
• Variable Expression: A variable expression is an expression that contains variables along with numbers and operation to define an expression. A few examples of a variable expression include 4x+y, 5ab + 33, etc.

### Some Algebraic Expression Formula

1. $$( m+p )^2 = m^2 + p^2 + 2m p$$
2. $$( m-p )^2 = m^2 + p^2 – 2mp$$
3. $$( m^2 – p^2 ) = (m+p) (m-p)$$
4. $$( m+p ) ^3 = m^3 + p^3 +3mp (m+p)$$
5. $$( m-p ) ^3 = m^3 – 3m^2p+3p^2m – p^3$$
6. $$( a+b+c ) ^2 = a^2 + b^2 + c^2+ 2ab + 2bc + 2ca$$
7. $$( a-b-c ) ^2 = a^2 + b^2 + c^2 -2ab – 2ca + 2bc$$
8. $$(a+b) ^ 4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3+ b^4$$
9. $$(a-b) ^ 4 = a^4 – 4a^3b + 6a^2b^2 – 4ab^3+ b^4$$
10. $$a^m \times a^n = a^ {(m+n)}$$
11. $$\frac{a^m}{a^n} = a^ {(m-n)}$$
12. $$p ^2 – q ^2 = (p-q) ^2 + 2 p q$$

## Solved Examples Algebraic Expression Formula

Q.1: Evaluate the $$(102)^2$$ using algebraic expressions.

Solution: As$$(102)^2$$ can be expressed as,

$$(102)^2 = (100 + 2)^2$$

Now, using the formula,

$$( m+p )^2 = m^2 + p^2 + 2m p$$

Thus,

$$(102) ^2 = 100 ^2 + 2 ^2 + 2 \times 100 \times 2$$

= 10000 + 4 + 400

= 10404

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