Algebra IOQM theory - 1

Common identities:

Common Algebraic Identities :

(i)
$(a + b)^2 = a^2 + 2ab + b^2$

(ii)
$(a - b)^2 = a^2 - 2ab + b^2$

(iii)
$a^2 - b^2 = (a - b)(a + b)$

(iv)
$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

(v)
$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

(vi)
$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

(vii)
$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

(viii)
$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

(ix)
$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$

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Let's spend some time on the last (ix) identity:

a + b + c = 0   ⇒   a² + b² + c² = 3abc
a³ + b³ + c³ = 3abc   ⇒   (a + b + c)(a² + b² + c² - ab - bc - ca) = 0

a³ + b³ + c³ = 3abc ⇒ (a + b + c)(a² + b² + c² - ab - bc - ca) = 0
=> a + b + c = 0 OR a² + b² + c² - ab - bc - ca = 0

a² + b² + c² - ab - bc - ca = 0 (multiply by 2)
=> 2a² + 2b² + 2c² - 2ab - 2bc - 2ca = 0
=> a² - 2ab + b² - 2bc + c² - 2ca + a² + b² + c² = 0
⇒ (a - b)² + (b - c)² + (c-a)² = 0
=> a = b = c
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(X) a² + b² + c² - ab - bc - ca = (1/2) [(a - b)² + (b - c)² + (c - a)²]

(XI) aⁿ - bⁿ = (a - b)(aⁿ⁻¹ + aⁿ⁻²b + aⁿ⁻³b² + ⋯ + abⁿ⁻² + bⁿ⁻¹)
a² - b² = (a - b)(a + b)
a³ - b³ = (a - b)(a² + ab + b²)
a⁴ - b⁴ = (a - b)(a³ + a²b + ab² + b³)

(XII)

$(x - a)(x - b) = x^2 - (a + b)x + ab$

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(XIII)
Let $n$ be an odd natural number:

$$a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}{b}^2+\cdots +b^{n-1})<br><br>$$

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Next intro to Complex Numbers.
Imaginary might be wrong word, they just exist in different dimension(Plane rather than 1 line dimension).
Multiplication/Addition/Division of Complex numbers.
Complex Conjugate and its use in division (similar to rationalization).
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Quadratic equation: Dharacharya formula, discriminant and its usages.
Roots are always 2, solutions may be 1 or 2.
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AP,GP,HP and their nth term.
Arithmetic,Geometric,Harmonic mean.