Algebraic Expressions - Formulas, Simplifying, Evaluating

In algebraic expressions, numbers are represented by letters or alphabets without their numerical values being specified. We learned how to represent an unknown value with letters like x, y, and z in algebraic basics. Here, we refer to these letters as variables. In an algebraic expression, both constants and variables can be included. Any value that is added before a variable and then multiplied by it is referred to as a coefficient. An algebraic expression is a mathematical statement that is generated when variables and constants are subjected to addition, subtraction, multiplication, division, and other mathematical operations.

What Is An Algebraic Expression?

A mathematical expression that includes variables, constants, and algebraic operations such as addition, subtraction, etc is known as an algebraic expression. Terms combine to form expressions. For instance, 2a + 3b - 8. Unknown variables, constants, and coefficients are used to represent this expression. An expression is the combination of these three terms. An algebraic expression does not have sides or an equal sign, unlike an algebraic equation.

Terms Associated With Algebraic Expressions

We operate with variables, symbols, and letters in algebra whose values are unknown to us. An algebraic expression is made up of various components. Consider an algebraic expression, 9x - 5.

Here, x is a variable, whose value is unknown to us and which may take any value.

9 is the coefficient of variable x because it is a well-defined constant value used with the variable. 5 is the constant that has a fixed numerical value.

Due to the two unlikely terms in the entire expression, it is known as the Binomial term.

Types Of Algebraic Expressions

Algebraic expressions can be divided into three main groups, including:

• Monomial expression - An algebraic expression with only one term is called a monomial. For instance, 6ab is a monomial expression.

• Binomial expression - A binomial expression is an algebraic expression that has two terms that are different from one another. For instance, 2xy + z is a binomial expression.

• Polynomial expression - A polynomial expression is one that contains many terms with non-negative integral exponents of a single variable. For instance, ax + by + c is a polynomial expression.

In addition to the monomial, binomial, and polynomial kinds, an algebraic expression can be classified into two further categories. These are:

• Numeric expression - A numeric expression is made up of numbers and operations; variables are never present. For instance, 10+8 is a numeric expression.

• Variable expression - An expression that uses variables, numbers, and an operation to define the expression is referred to as a variable expression. For instance, 5ab + 10 is a variable expression.

Formulas to solve algebraic expressions

To evaluate the expressions or equations, we generally utilize the following algebraic formulas:

• (a+b)^{2}=a^{2}+2ab+b^{2}

• (a-b)^{2}=a^{2}-2ab+b^{2}

• a^{2}-b^{2}=(a+b)(a-b)

• (a+b)^{3}=a^{3}+b^{3}+3ab(a+b)

• (a-b)^{3}=a^{3}-b^{3}-3ab(a-b)

• a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})

• a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})

How To Simplify Algebraic Expressions?

Simply combining like terms will simplify an algebraic expression. As a result, similar variables will be combined. Now, the identical powers will be merged from similar variables.

• Adding algebraic expressions - Add the terms having variables with similar powers. For instance, (5x^{2}-6x)+(x^{2}+8x-2)=(5x^{2}+x^{2})+(-6x+8x)-2

(5x^{2}-6x)+(x^{2}+8x-2)=6x^{2}+2x-2

• Subtracting algebraic expressions - Adding the additive inverse of the second expression to the first expression results in the subtraction of two algebraic expressions. For instance, (5x^{2}-6x)-(x^{2}+8x-2)=(5x^{2}-6x)+(-x^{2}-8x+2)

(5x^{2}-6x)-(x^{2}+8x-2)=(5x^{2}-x^{2})+(-6x-8x)+2

(5x^{2}-6x)-(x^{2}+8x-2)=4x^{2}-14x+2

• Multiplying algebraic expressions - We multiply each term in the first expression by each term in the second expression to multiply two algebraic expressions, and then we add up all the products. For instance,

(3ab+2)(2ab+9)=(3ab)(2ab)+9(3ab)+2(2ab)+2(9)

(3ab+2)(2ab+9)=6a^{2}b^{2}+31ab+18

• Dividing algebraic expressions - We factor the numerator and denominator, eliminate any possible terms, and then simplify the remaining terms to divide two algebraic expressions. For instance, \frac{3x^{2}}{(9x+18)}=\frac{3x^{2}}{9(x+2)}=\frac{x^{2}}{3(x+2)}