Algebraic expressions: A collection of constants and variables connected one or more of the operations of addition, subtraction, multiplication and division are called an algebraic expression. The various parts of an algebraic expression are separated by signs ‘+’, ‘-’, ‘/’, ‘x’ is called terms of algebraic expression.

An algebraic expression can contain one term (monomial), two terms (binomial), three terms (trinomial) and more than three terms (multinomial).

Constants: It is a symbol which has a fixed value.

Variables: Variables are the symbols which can be given various numerical values.

Factors: Each of the quantities multiplied together to form a product is called a factor of the product.

Coefficient: Any factor of a (non- constant) term of an algebraic expression is called the coefficient of the remaining factor of the term. It can be of two types- numerical coefficient and literal coefficient.

Multinomials are also called polynomials. It is the algebraic expressions containing two or more variables such that the power of the variables in each term are non-negative integers.

Take the sum of the powers of variables in each term; the greatest sum is the degree of the polynomial.

Linear Polynomial - A polynomial of degree one.

Quadratic Polynomial - A polynomial of degree two.

Cubic Polynomial - A polynomial of degree three.

Algebraic identities: An algebraic identity is an equality that holds for any values of its variables. Since an identity holds for all values of its variables, it is attainable to substitute instances of one side of the equality with the other side of the equality.

Smart use of identities offers shortcuts to many problems by making the algebra easier to manipulate. Below are lists of some common algebraic identities.

The algebraic expressions and identities class 8 formulas given below are the extraction of law of product and expansion of algebraic expressions that follow distributive law.

a(b+c) = ab + bc

(a+b)c = ac + bc

In this article, algebraic expressions and identities class 8 formulas we are going to deal with the products and expansion of the forms - (x±a) (x±b), (x±a)^{2}.

(a+b)^{2}= a^{2}+ 2ab +b^{2}

(a-b)^{2}= a^{2}- 2ab +b^{2}

(a+b)(a-b)= a^{2}-b^{2}

(x+a)(x+b)= x^{2}+ (a+b)x+ab

(x+a)(x-b)= x^{2}+ (a-b)x-ab

(x-a)(x+b)= x^{2}+ (b-a)x-ab

(x-a)(x-b)= x^{2}- (a+b)x+ab

(a+b)^{3}= a^{3}+ 3ab(a+b) +b^{3}

(a-b)^{3}= a^{3}- 3ab(a-b) -b^{3}

1. Find the following products:

(x+2) (x+5)

(x+6) (x-4)

(x-3) (x+7)

Solution:

(x+2) (x+5)

Thus, to solve this we can use the formula (x+a)(x+b)= x^{2}+ (a+b)x+ab

Where a = 2 and b = 5.

On replacing the value,

(x+a)(x+b)= x^{2}+ (a+b)x+ab

(x+2) (x+5) = x^{2}+ (2+5)x+10

(x+2) (x+5) = x^{2}+ (7)x+10

Therefore, (x+2)(x+5)is equal to x^{2}+ 7x+10

(x+6) (x-4)

To solve this we can use the formula (x+a)(x-b)= x^{2}+ (a-b)x-ab

Where a = 6 and b = 4.

On replacing the value,

(x+a)(x-b)= x^{2}+ (a-b)x-ab

(x+6)(x-4)= x^{2}+ (6-4)x-24

(x+6)(x-4)= x^{2}+ 2x-24

Therefore, (x+6)(x-4)is equal to x^{2}+ 2x-24.

(x-3) (x+7)

To solve this we can use the formula (x-a)(x+b)= x^{2}+ (b-a)x-ab

Where a = 3 and b = 7.

On replacing the value,

(x-a)(x+b)= x^{2}+ (b-a)x-ab

(x-3)(x+7)= x^{2}+ (7-3)x-21

(x-3)(x+7)= x^{2}+ 4x-24

Therefore, (x-3)(x+7)is equal to x^{2}+ 4x-24.

FAQ (Frequently Asked Questions)

Question 1: What are the Operations of Algebraic Expressions?

Answer: As we have studied before, the terms of algebraic expressions are separated by signs ‘+’, ‘-’, ‘/’, ‘x’. These signs make up the algebraic expression but the operations like addition, subtraction, multiplication and division of algebraic expression is done of two or more algebraic expressions. There are four operations on algebraic expression:

Addition: The sum of two or more like terms is a like term whose coefficient is the sum of coefficients of like terms.

Subtraction: For subtraction, change the sign of the term to be subtracted and then add.

Multiplication: Multiply each term of the polynomial by a monomial. Product of two monomials = (product of numerical coefficient) x (product of variable coefficients).

- Division: Divide each term of the polynomial by the monomial. Quotient of two monomials = (quotient of numerical coefficient) x (the quotient of variable coefficients) .

Question 2: How to Divide a Polynomial by Another Polynomial?

Answer: For dividing a polynomial by another polynomial, proceed according to the following steps:

i) First, arrange the terms of the dividend and divisor in descending order of their degrees.

ii) Divide the first term of the dividend by the first term of the divisor to achieve the first term of the quotient.

iii) Then, multiply all the terms of the divisor by the first term of the quotient and subtract the result from the dividend.

iv) Consider the remainder as a new dividend and proceed as before.

v) Repeat the process until we obtain a remainder which is either zero or a polynomial of degree less than that of the divisor.