Algebra Basics and Formulae

What is Algebra?

Algebra is a part of mathematics where we use letters/symbols to represent numbers in an equation/formula/inequality, and then manipulate them using some rules.

Symmetrical and Unsymmetrical expression

Unsymmetrical expression: It is an expression in which the weight of all the variables (say a, b,c …) is not the same.

For example, $a + b + c^{2}, a^{2} + b^{2} - c^{2}, a^{3} + b^{3} + c^{2}$ etc
Symmetrical expression: It is an expression in which the weight of all the variables is equal.

For example, a + b + c, 3a + 3b + 3c, $a^{2} + b^{2} + c^{2}, a^{3} + b^{3} + c^{3}$ , ab + bc + ca etc.

So, in a symmetrical expression we can interchange the positions of the variables.

Algebra Formulae

Now, let’s have a look at some of the basic formulae that we are going to use in this chapter.

Square Formulae

Square Formulae Type 1: Two Variables

$(x + y)^{2} = x^{2} + y^{2} + 2 x y = (x - y)^{2} + 4 x y$
$(x - y)^{2} = x^{2} + y^{2} - 2 x y = (x + y)^{2} - 4 x y$

$x^{2} - y^{2} = (x + y) (x - y)$
$x^{2} + y^{2} = (x + y)^{2} - 2 x y = (x - y)^{2} + 2 x y$

$(x + y)^{2} + (x - y)^{2} = 2 (x^{2} + y^{2})$
$(x + y)^{2} - (x - y)^{2} = 4 x y$

Square Formulae Type 2: Three Variables

$(x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2 (x y + y z + z x)$
$(x - y + z)^{2} = x^{2} + y^{2} + z^{2} + 2 (- x y - y z + z x)$
$(x - y - z)^{2} = x^{2} + y^{2} + z^{2} + 2 (- x y + y z - z x)$

Square Formulae Type 3

If $(x - a)^{2} + (y - b)^{2} + (z - c)^{2}$ = 0, then x – a = 0, y – b = 0 & z – c = 0
i.e. If $A^{2} + B^{2} + C^{2}$ = 0, then A = B = C = 0

$A^{2} + B^{2} + C^{2} - A B - B C - C A$ = 0, then A = B = C

Cubic Formulae

Cubic Formulae Type 1: Two Variables

$(x + y)^{3} = x^{3} + y^{3} + 3 x y (x + y)$
$(x - y)^{3} = x^{3} - y^{3} - 3 x y (x - y)$

$(x + y)^{3} + (x - y)^{3} = 2 x (x^{2} + 3 y^{2})$
$(x + y)^{3} - (x - y)^{3} = 2 y (3 x^{2} + y^{2})$

$x^{3} + y^{3} = (x + y) (x^{2} + y^{2} - x y) = (x + y)^{3} - 3 x y (x + y)$
$x^{3} - y^{3} = (x - y) (x^{2} + y^{2} + x y) = (x - y)^{3} + 3 x y (x - y)$

x^{n} - y^{n} = (x - y) (x^{n - 1} + x^{n - 2} . y + x^{n - 3} . y^{2} + \dots + y^{n - 1})

Cubic Formulae Type 2: Three Variables

$(x + y + z)^{3} = x^{3} + y^{3} + z^{3} + 3 (x + y) (y + z) (z + x)$

$x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)$
= $(x + y + z) \times \frac{1}{2} [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]$

When x + y + z = 0, then

x^{3} + y^{3} + z^{3} = 3 x y z

Function and Inverse Function Formulae

Type 1: $x + \frac{1}{x}$

If $x + \frac{1}{x}$ = a, then:

$x - \frac{1}{x} = \sqrt{a^{2} - 4}$
$x^{2} + \frac{1}{x^{2}} = a^{2} - 2$
$x^{3} + \frac{1}{x^{3}} = a^{3} - 3 a$

If $x + \frac{1}{x}$ = 1, then $x^{3}$ = –1.
If $x + \frac{1}{x}$ = -1, then $x^{3}$ = 1

If $x + \frac{1}{x}$ = 2, then x = 1
If $x + \frac{1}{x}$ = -2, then x = – 1

If $x + \frac{1}{x} = \sqrt{3}$ , then $x^{6}$ = –1

Type 2: $x - \frac{1}{x}$

If $x - \frac{1}{x}$ = a, then:

$x + \frac{1}{x} = \sqrt{a^{2} + 4}$
$x^{2} + \frac{1}{x^{2}} = a^{2} + 2$
$x^{3} - \frac{1}{x^{3}} = a^{3} + 3 a$

Type 3: $x^{2} + \frac{1}{x^{2}}$

If $x^{2} + \frac{1}{x^{2}}$ = a, then:

$x + \frac{1}{x}$ = $\sqrt{a + 2}$
$x - \frac{1}{x}$ = $\sqrt{a - 2}$

If $x^{2} + \frac{1}{x^{2}}$ = 1, then:

$x^{3} + \frac{1}{x^{3}}$ = 0 and $x^{6}$ = –1

Type 4: Higher Even Powers

$x^{4} + \frac{1}{x^{4}} = (x^{2} + \frac{1}{x^{2}})^{2} - 2 = [(x - \frac{1}{x})^{2} + 2]^{2} - 2$

$x^{6} + \frac{1}{x^{6}} = (x^{3} + \frac{1}{x^{3}})^{2} - 2 = (x^{2} + \frac{1}{x^{2}})^{3} - 3 (x^{2} + \frac{1}{x^{2}})$

Type 5: Higher Odd Powers

$x^{5} + \frac{1}{x^{5}} = (x^{2} + \frac{1}{x^{2}}) (x^{3} + \frac{1}{x^{3}}) - (x + \frac{1}{x})$

$x^{7} + \frac{1}{x^{7}} = (x^{4} + \frac{1}{x^{4}}) (x^{3} + \frac{1}{x^{3}}) - (x + \frac{1}{x})$

$x^{5} - \frac{1}{x^{5}} = (x^{2} + \frac{1}{x^{2}}) (x^{3} - \frac{1}{x^{3}}) - (x - \frac{1}{x})$

$x^{7} - \frac{1}{x^{7}} = (x^{4} + \frac{1}{x^{4}}) (x^{3} - \frac{1}{x^{3}}) + (x - \frac{1}{x})$

Algebra Properties

Property 1: Distributive law

x (y + z) = xy + xz

Property 2: Componendo and Dividendo

If $\frac{a}{b} = \frac{x}{y}$ , then $\frac{a + b}{a - b} = \frac{x + y}{x - y}$

If $\frac{x + y}{x - y}$ = z, then , $\frac{x}{y} = \frac{z + 1}{z - 1}$

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Last updated on Aug 10, 2021