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Updated on 02nd March, 2023 , 3 min read
Mathematicians often use a set of formulas known as algebraic identities. They serve as the basis for algebra and make computations simple and straightforward. To solve some algebraic problems, one must perform a number of complicated mathematical operations. Here, we can perform the calculations directly using algebraic identities, skipping any additional steps.
An algebraic identity states that, for all possible values of the variables, the left side of the equation equals the right side. Here, we'll try to familiarize ourselves with all of the algebraic identities, their justifications, and how to apply them to mathematical calculations.
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Algebraic expressions are categorized as monomials, binomials, or trinomials depending on how many terms, one, two, or three, are present. Additionally, an expression is referred to as a polynomial if it contains one or more terms. A number called a coefficient is associated with each term in an algebraic expression.
1. (a+b)2=a2+2ab+b2
2. (a-b)2=a2-2ab+b2
3. (a+b) (a-b) = a2- b2
If values are entered for terms a and b in any one of the three expressions listed above, the left side of the equation will be equal to the right side of the equation. demonstrating these expressions' identities.
The following are the identities in algebra with two variables. These identities are easily verified by multiplying polynomials and expanding the square or cube. For example, we can demonstrate that (a + b)2 = (a + b) (a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2 to demonstrate the first identity below. The other identities can be confirmed in the same manner.
The identities for three variables in algebra have also been derived, just like the identities for two variables. Additionally, these identities make it simple to work with algebraic expressions by requiring the fewest steps possible.
Algebraic identities are very helpful in quickly factoring algebraic expressions. Some of the higher algebraic expressions, like a4 - b4, can be factored using the fundamental algebraic identities like a2 - b2 = (a - b)(a + b). The following is a list of algebraic identities that can be used to factor polynomials.
If x + y = 5 and xy = 6 and x > y, then find 2(x2 + y2).
Solution :
(x + y)2 = (x + y)(x + y)
(x + y)2 = x2 + xy + xy + y2
(x + y)2 = x2 + 2xy + y2
or
x2 + 2xy + y2 = (x + y)2
Subtract 2xy from both sides.
x2 + y2 = (x + y)2 - 2xy
Substitute x + y = 5 and xy = 6.
x2 + y2 = 52 - 2(6)
x2 + y2 = 25 - 12
x2 + y2 = 13
Multiply both sides by 2.
2(x2 + y2) = 2(13)
2(x2 + y2) = 26
If a3 - b3 = 513 and a - b = 3, what is the value of ab?
Solution :
a3 - b3 = 513
(a - b)3 + 3ab(a - b) = 513
Substitute a - b = 3.
33 + 3ab(3) = 513
27 + 9ab = 513
Subtract 27 from both sides.
9ab = 486
Divide both sides by 9.
ab = 54
If a + (1/a) = 11, then determine the value of a2 + 1/a2.
Solution:
Given algebraic equation: a + (1/a) = 11
To find: a2 + 1/a2.
Using the algebraic identity, (a+b)2 = a2 + b2 + 2ab.
Thus, (a + 1/a )2 = a2 + (1/a)2 + 2(a)(1/a)
Now, substitute a + (1/a) = 11 in the above equation, we get
(11)2 = a2 + 1/a2 + 2
121 = a2 + 1/a2 + 2
a2 + 1/a2 = 121 – 2
a2 + 1/a2 = 119.
Therefore, the value of a2 + 1/a2 = 119.
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Two simple methods can be used to verify algebraic identification. Substitution math is one approach, where we change the value in the algebraic identifier to find the variable. There are numerous expressions for algebraic identities on both sides of the signed equation. Here, we can try to get the same result on both sides of the equation by substituting the values on the two sides of the equation. A different approach is to use the algebraic solution to obtain the right side of the equation by optimizing and simplifying the left side of the equation. For this method to work, one needs to be familiar with geometric concepts and specific materials
Three significant identities make up the class 8 algebraic identities. Here is a list of them: (a+b)2 = a2+2ab+b2(a-b)2 = a2- 2ab+b2a2-b2= (a+b) (a-b)
No, not all equations are regarded as identities. However, we can assert that each identity is an equation.
Numerous applications of the algebraic identities exist across the board in mathematics. These algebra identities serve as the foundation for the entire subject of algebra. Algebraic identities are also widely used in areas of study like geometry, coordinate geometry, trigonometry, and calculus. Finding these is beneficial for finding straightforward solutions to problems.
Algebra is a mathematical concept that combines both numbers and letters. In the algebraic formula, the numbers are fixed because their value is known, and the unknown quantities that need to be determined are represented by the letters or alphabets.
When we first began studying math as students, numbers were everything. Integers, natural numbers, and more integers Then we began to study mathematical operations such as addition, subtraction, BODMAS, and so forth. The eighth grade introduces alphabets and letters to the curriculum. This was the start of our introduction to algebra. Algebra is a branch of mathematics that combines both numbers and letters. The letters or alphabet in algebraic formulas stand in for the unknown number while the numbers remain constant or have a known value
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