Algebra - Introduction and Basic Formula
INTRODUCTION AND BASIC FORMULAS IN ALGEBRA
Algebra is the part of mathematics that helps to represent the problems or situations in the form of mathematical expressions. Algebraic formula are used to simplify the algebraic statement. Algebraic formulas are useful for resolving algebraic, quadratic, polynomials, trigonometry, probability, and more.
Algebra is the branch of mathematics in which arithmetical operations and formal manipulations are applied to abstract symbols rather than specific numbers. The notion that there exists such a distinct subdiscipline of mathematics, as well as the term algebra to denote it, resulted from a slow historical development. This article presents that history, tracing the evolution over time of the concept of the equation, number systems, symbols for conveying and manipulating mathematical statements, and the modern abstract structural view of algebra. For information on specific branches of algebra, see elementary algebra, linear algebra and modern algebra.
Algebraic formulas are the combination of numbers and letters to form an equation or formula. The algebraic formula is a short quick formula to solve complex algebraic calculations.
The properties of algebra enable us to solve mathematical equations. Notice that these properties hold for addition and multiplication. These properties include the associative property, commutative property, distributive property, identity property, inverse property, reflexive property, symmetric property, and transitive property.
a2 – b2 = (a – b)(a + b)
(a + b)2Â = a2Â + 2ab + b2
a2 + b2 = (a + b)2 – 2ab
(a – b)2 = a2 – 2ab + b2
(a + b + c)2Â = a2Â + b2Â + c2Â + 2ab + 2bc + 2ca
(a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
(a + b)3Â = a3Â + 3a2b + 3ab2Â + b3Â
( a + b )3Â = a3Â + b3Â + 3ab(a + b)
(a – b)3 = a3 – 3a2b + 3ab2 – b3 or a3 – b3 – 3ab(a – b)
a3 – b3 = (a – b)(a2 + ab + b2)
a3 + b3 = (a + b)(a2 – ab + b2)
(a + b)4Â = a4Â + 4a3b + 6a2b2Â + 4ab3Â + b4
(a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4
a4 – b4 = (a – b)(a + b)(a2 + b2)
a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
If n is a natural number an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +an-3b2…- bn-2a + bn-1)
(a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….)
Laws of Exponents (am)(an) = am+n ; (ab)m = ambm ; (am)n = amn
Changing the order of addons does not change the sum.
Multiplication : a x b = b x a
Changing the order of the factor does not change the product.
Addition : (a + b)+ c = a + (b + c)
Changing the grouping of the addends does not change the sum.
Multiplication : (a x b) xc = a x (b x c)
Changing the grouping of the factors does not change the product.
Distributive properties:Â
Addition :Â a Ă— (b + c) = a Ă— b + a Ă— c
Multiplication : Â (a + b) Ă— c = a Ă— c + b Ă— c
The distributive property states that multiplying each element by a single term and then adding and subtracting the products is the same as multiplying each component by a single term and then adding and subtracting the products.
Rule of multiplication over subtraction: p (q-r) = p*q – p*rÂ
If p, q, and r, are all integers.
Left distributive law if p* (q-r) = (p * q) – (p*r)- and
Right distributive law if (p-q)*r = (p*r) – (q*r)-
What are the properties of Algebra?
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