Algebra Notes

Algebra

Standard algebraic identities and the x + 1/x family; factorisation and the factor and remainder theorems; linear equations in one and two variables and when a pair has no solution or infinitely many; quadratic equations — solving, nature of roots from the discriminant, sum and product of roots, forming an equation from its roots; arithmetic and geometric progressions with their nth term and sums, including an infinite GP; simple inequalities — with fully worked examples.

Last reviewed 22 Sept 2026 · 6 min read

Identities worth knowing by sight

FormulaStandard identities

(a \pm b)^{2} = a^{2} \pm 2 ab + b^{2}, a^{2} - b^{2} = (a + b) (a - b)

(a + b)^{3} = a^{3} + b^{3} + 3 ab (a + b), (a - b)^{3} = a^{3} - b^{3} - 3 ab (a - b)

a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2}), a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})

(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (ab + b c + c a)

a^{3} + b^{3} + c^{3} - 3 ab c = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a)

If $a + b + c = 0$ , the last line gives $a^{3} + b^{3} + c^{3} = 3 ab c$ .

FormulaThe x + 1/x family

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

x^{2} + \frac{1}{x ^{2}} = k^{2} - 2, x^{3} + \frac{1}{x ^{3}} = k^{3} - 3 k, x - \frac{1}{x} = \pm k^{2} - 4

and if $x - \frac{1}{x} = k$ , then $x^{2} + \frac{1}{x ^{2}} = k^{2} + 2$ .

Polynomials

Remainder theorem: the remainder when $p (x)$ is divided by $(x - a)$ is $p (a)$ .
Factor theorem: $(x - a)$ is a factor exactly when $p (a) = 0$ .
A cubic is factorised by finding one root among the divisors of the constant term, dividing, and factorising the quadratic that is left.

Linear equations

For $a_{1} x + b_{1} y = c_{1}$ and $a_{2} x + b_{2} y = c_{2}$ :

Condition	Meaning
$\frac{a _{1}}{a _{2}} \neq = \frac{b _{1}}{b _{2}}$	One unique solution (lines cross)
$\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}}$	No solution (parallel lines)
$\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}$	Infinitely many (the same line)

Quadratic equations

FormulaRoots of

a x^{2} + b x + c = 0

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}, D = b^{2} - 4 a c

$D > 0$ : two distinct real roots ( $D$ a perfect square → rational)
$D = 0$ : two equal real roots
$D < 0$ : no real roots
Sum of roots $= - \frac{b}{a}$ , product $= \frac{c}{a}$
An equation with roots $α, β$ is $x^{2} - (α + β) x + α β = 0$

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