Algebraic Identities
Nov 30, 2022, 16:45 IST
Algebraic identities is an important set of formulas in mathematics. They form the basis of algebra and are useful in performing calculations in simple and easy steps. Certain algebra problems require working through many mathematical steps to get the answer. Here, using algebraic identities, we can perform calculations without further steps.
Algebraic identity means that the left side of the equation is identically equal to the right side for all values of the variables. Here we will try to get acquainted with all the algebraic identities, their proofs, and how to use these identities in our mathematical calculations.
Algebraic identities are equations in algebra in which the value of the left side of the equation is equal to the value of the right side of the equation. They are satisfied with any variable values. Let's take a few examples to understand this better. Consider the equations: 6x - 6 = 12, 10x - 6 = 24. These equations only work for a certain value(s) of x and generally do not work for any value. Now consider the equation x
2
- 16 = (x + 4)(x - 4). Always remember that this equation holds for any value of x (try substituting any number for x on both left and the right sides, you should get the same answer).
These are helpful in various mathematical problems. Some four basic algebra identities are as follows.
-
(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
+ x(a + b) + ab
All Algebraic Identities have been derived from the Binomial Theorem, which is given as:
(a+b)
n
=
n
C
0
.a
n
.b
0
+
n
C
1
.a
n−1
.b
1
+……..+
n
C
n−1
.a
1
.b
n−1
+
n
C
n
.a
0
.b
n
Some of the Standard Algebraic Identities listed are mentioned below:
-
Identity I
: (a + b)
2
= a
2
+ 2ab + b
2
-
Identity II
: (a – b)
2
= a
2
– 2ab + b
2
-
Identity III
: a
2
– b
2
= (a + b)(a – b)
-
Identity IV:
(x + a)(x + b) = x
2
+ (a + b) x + ab
-
Identity V
: (a + b + c)
2
= a
2
+ b
2
+ c
2
+ 2ab + 2bc + 2ca
-
Identity VI
: (a + b)
3
= a
3
+ b
3
+ 3ab (a + b)
-
Identity VII:
(a – b)
3
= a
3
– b
3
– 3ab (a – b)
-
Identity VIII:
a
3
+ b
3
+ c
3
– 3abc = (a + b + c)(a
2
+ b
2
+ c
2
– ab – bc – ca)
Following are the identities in algebra with two variables. Expanding the square/cubic algebraic Identities and performing polynomial multiplication can easily verify these identities. For example, to verify the first identity below, (a + b)
2
= (a + b) (a + b) = a
2
+ ab + ab + b
2
= a
2
+ 2ab + b
2
. We can verify other identities in the same way.
Example: Expand (3x + y)
2
Ans.
To expand the given expression, substitute a = 3x and b = y in (a + b)
2
= a
2
+ 2ab + b
2
,
(3x + y)
2
= (3x)
2
+ 2(3x)(y) + y
2
= 9x
2
+ 6xy + y
2
The algebra identities for the three variables were also derived exactly as the two variables were. These identities further help to easily work with algebraic expressions with the least number of steps.
-
(a + b + c)
2
= a
2
+ b
2
+ c
2
+ 2ab + 2bc + 2ac
-
a
2
+ b
2
+ c
2
= (a + b + c)
2
- 2(ab + bc + ac)
-
a
3
+ b
3
+ c
3
- 3abc = (a + b + c)(a
2
+ b
2
+ c
2
- ab - ca - bc)
-
(a + b)(b + c)(c + a) = (a + b + c)(ab + ac + bc) - 2abc
Example: When a + b + c = 0, then find the value of a
3
+ b
3
+ c
3
?
Ans.
By one of the above identities,
a
3
+ b
3
+ c
3
- 3abc = (a + b + c)(a
2
+ b
2
+ c
2
- ab - ca - bc)
Substituting (a + b + c) = 0, we get
a
3
+ b
3
+ c
3
- 3abc = 0 (a
2
+ b
2
+ c
2
- ab - ca - bc)
a
3
+ b
3
+ c
3
- 3abc = 0
a
3
+ b
3
+ c
3
= 3abc
Algebraic identities are very useful in easily factoring algebraic expressions. Using these identities, some higher algebraic expressions like a4 - b4 can be easily factored using basic algebraic identities like a2 - b2 = (a - b)(a + b). The list below is a set of algebraic identities valid for factorization polynomials.
-
a
2
- b
2
= (a - b)(a + b)
-
x
2
+ x(a + b) + ab = (x + a)(x + b)
-
a
3
- b
3
= (a - b)(a
2
+ ab + b
2
)
-
a
3
+ b
3
= (a + b)(a
2
- ab + b
2
)
Example: a
4
- b
4
Ans.
(a
2
)
2
- (b
2
)
2
= (a
2
- b
2
) (a
2
+ b
2
)
= (a - b)(a + b)(a
2
+ b
2
)
Q1. Find the product of (x + 2)(x + 2) using standard algebraic identities.
Ans.
(x + 2)(x + 2) can be written as (x + 2)
2
. Thus, it is of Identity I where a = x and b = 2. So we have,
(x + 2)
2
= (x)
2
+ 2(x)(2) + (2)
2
= x
2
+ 4x + 4
Q2. Using identities, solve 296 × 304.
Ans.
296 × 304 can be written as ( 300 - 4 ) × ( 300 + 4)
And this is based on the algebraic identity (a + b)(a - b) = a
2
- b
2
Here we have a = 300, and b = 4
Substituting the values in the above identity, we get:
(300 - 4)(300 + 4) = 300
2
- 4
2
= 90000 - 16
= 89984
Q3. Factorise (x
4
– 1) by using standard algebraic identities.
Ans.
(x
4
– 1) is of the form Identity III where a = x
2
and b = 1. So we have,
(x
4
– 1) = ((x
2
)
2
– 1
2
) = (x
2
+ 1)(x
2
– 1)
The factor (x
2
– 1) can be further factorized using the same Identity III where a = x and b = 1.
So, (x
4
– 1) = (x
2
+ 1)((x)
2
–(1)
2
)
= (x
2
+ 1)(x + 1)(x – 1)
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Q1. Are algebraic identities used in daily life?
Ans.
Algebraic identities are the standard identities that are used to simplify complex equations. The two types of algebraic identities are Binomial and trinomial Algebraic Identities. These identities are used in real life.
Q2. Why are algebraic identities important?
Ans.
Recognizing and knowing these identities help the students to understand mathematical procedures. It will also help them to develop fluency when applying these procedures in algebraic manipulations and problem-solving.
Q3. Who invented algebraic identities?
Ans.
Muhammad ibn Musa al-Khwarizmi invented algebric identities. He was a 9th-century Muslim mathematician and astronomer. He is also known as “
father of algebra
,” a word derived from the title of his book, Kitab al-Jabr.
Q4. Write three algebraic identities in Maths.
Ans.
The three algebraic identities in Maths are-
-
(a+b)
2
= a
2
+ b
2
+ 2ab
-
(a-b)
2
= a
2
+ b
2
– 2ab
-
a
2
– b
2
= (a+b) (a-b)
Q5. What is the use of algebraic identities?
Ans.
Algebraic identities are used to solve any algebraic expression or polynomial faster. It makes the calculation easier.
