Algebra Formulas for class 12

Algebra Formula Class 12

What is algebra in Mathematics?

  • Algebra is one of the branches of Mathematics which deals with variables and numbers.

  • In Mathematics, algebra is one of the most important topics in which general symbols and letters represent quantities and numbers in equations and formulae.

  •  Elementary algebra is the more basic parts of Algebra and more abstract parts are known as modern algebra or abstract algebra. 

  • It is very important as it covers everything from elementary equation solving to the study of abstractions such as rings, groups, and fields.

What is an algebraic expression in Mathematics?

  • An algebraic expression is a combination of constants and variables that are connected by addition, subtraction, multiplication, and division signs.

  • Various parts of an algebraic expression that are separated by the signs of addition (+) or subtraction (-) are known as terms of the algebraic expression.

  • An algebraic expression can be defined as a sum, difference, product, or quotient of constants and variables.

  • Let us take an example,

 2x + 4 

The above expression is known as an algebraic expression where the value 

of x may vary, as x is a variable, and 4 is a constant. 2x and 4 are the terms of an algebraic expression that are separated by an addition (+) sign. Anything (a,b, c…….z) can be written in place of x as they all are variables.


Methods to verify algebraic identities:

  1. Using the Substitution Method.

  • Substitution generally means putting numbers or values in the place of variables or letters.

  • In the substitution method, an arithmetic operation is performed by substituting the values for the variables.

  • For example, when we have x-2=4

When we substitute x= 6, 

On the Right- hand side,

4

On the left-hand-side,

x-2 = 6-2 = 3

Here, Right-hand side = Left-hand side which means (x-2) is an identity.


Suppose, (a+3) (a-3) = (a2-9) 

Substituting a= 1

On the Right- hand side,

(a2-9) = (1-9) = -8

On the Left- hand side,

(a+3) (a-3) = (1+3) (1-3) = (4) (-2) = -8

Here, Right-hand side = Left-hand side which means that (a+3) (a-3) is an identity.


  1. Using Activity Method.

  • In this method, the algebraic identity is verified geometrically by taking different values of x and y.

  • In the activity method, the identities are verified by cutting and pasting paper.

  • To verify identity using this method, you need to have a basic knowledge of Geometry.

           

Standard identities in Mathematics you need to know!

Identity I

(a+b)2 = a2+2ab+b2


Identity II

(a-b)2 = a2- 2ab+b2

Identity III

a2-b2= (a+b) (a-b)

Identity IV

(x+a) (x+b) = x2+(a+b) x+ab

Identity V

(a+b+c)2= a2+b2+c2+ 2ab+2bc+2ca

Identity VI

(a+b)3= a3+b3+3ab(a+b)

Identity VII

(a-b)3= a3 -b3-3ab(a-b)

Identity VIII

a3 +b3+c3-3abc


List of Algebra Formulas Class 12:

Here is the list of Algebra Formula for class 12 that will help students to guide them in their study of algebra. Basic formulas of Algebra:

  1. Formulas for powers-

pm x pn = pm+n

 

        {pm}⁄{pn} = pm-n


          (pm)n = pmn


p-m = 1/pm


p1 = p

P0 = 1

 


  1. Law of Exponents – 


                  (xm)(xn) = x m+n


            

                     (xy)m = xmym


          

                       (xm)n = xmn

 

  1. Polynomial formulas-

 

an – bn = (a – b)(an-1 + an-2 +…+ bn-2a + bn-1) where  n is a natural number.


(n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1) where n is an even number.


(n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +…- bn-2a + bn-1) where n is an odd number


 

(a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….)


Vector Algebra is included in the Class 12 mathematics syllabus. It generally deals with vectors – things that have both directions and magnitudes. Learning vector algebra will help you in handling geometric transformations and it also is very important in understanding Linear Algebra. Vectors hold great importance in both Physics and Mathematics.

If \[\overrightarrow a  = x\overrightarrow i  + y\overrightarrow j  + z\overrightarrow k \] then magnitude or the length or norm or absolute value of vector \[\overrightarrow a {\text{ }}is||\overrightarrow a ||a = \sqrt {{x^2} + {y^2} + {z^2}} \]


If  \[a = cos\alpha ,b = cos\beta ,c = cos\gamma ,{\text{ }}then{\text{ }}\alpha ,\beta ,\gamma ,\]are known as directional angles of the vectors and the value of  \[\overrightarrow a  = co{s^2}\alpha  + co{s^2}\beta  + co{s^2}\gamma  = 1\]


12th Algebra Formulas

12th Algebra Formulas are of foremost importance at every level of competitive exams. You should memorize these formulas to crack all the Math competitive exams.

The main objective of solving the algebraic equation is to find the unknown variables in the expression.


Fun facts: 

  • It was Babylonians who came up with Algebra in 1900 BC.

  • The use of signs addition (+) and subtraction (-) prove to be beneficial in performing algebraic equations. Before that, people use written words to express the functions of addition and subtraction which was a time-consuming process.




Questions to be solved:

Question 1)Expand the following.

                                 (a + 2)(a-2)


Solution)  (a + 2) (a- 2)

Let’s multiply the given expression,

= We can simplify the question by using the formula, a2 - b2

= (a)2 – (2)2

= a2 – 4 


FAQ (Frequently Asked Questions)

What is the Algebraic Equation?

An equation which is the combination of constants and variables connected by the signs of +,-, / or x is equal to another equation is called an algebraic equation.

A mathematical statement is said to be an algebraic equation if the left-hand side equation is equal to the right-hand side equation. That means values to both sides of equal to sign are equal.


Consider an equation 5 + 2 = 3 + 4 here you can see both sides have the value 6. Such expressions are called algebraic equations.


What is the difference between the algebraic equation and algebraic expression?

Students are often confused about the term Algebraic Equation and Algebraic expression. Here are the differences between Algebraic Equation and Algebraic expression- 

  • An algebraic expression is a single equation with no equality sign, whereas the Algebraic equation is a set of two equations that have equality sign between them.

  • An algebraic expression is of form 2x + 3y - x + y. 

  • An Algebraic equation is of the form 5x + 2 = 0.

  • An Algebraic expression can be expanded or simply whereas, an algebraic equation has to be solved, it has a certain value.

  • The algebraic expression will be true for any value of x, while the algebraic equation will be true only for some values of x.


What is a Vector in Math?

A vector in Mathematics can be defined as an object that has both a direction and a magnitude. Geometrically, a vector can be represented as a directed line segment, whose length is the magnitude of the vector and with an arrow that indicates the direction of the vector. Force and velocity are two examples of vectors.

How do vectors work?

The lines that represent both direction and magnitude (size) are known as vectors. If any object moves in more than one direction or more than one force acts upon any object equivalently, then the vectors can be added to find a resultant displacement or the resultant force that acts on the object.