Algebra
Identities:
- If $a+b+c=0, a^{3}+b^{3}+c^{3}=3 a b c$
- If $a+b+c=0, a^{4}+b^{4}+c^{4}=\frac{1}{2}\left(a^{2}+b^{2}+c^{2}\right)^{2}$
- Sophie Germain Identity: $ a^4 + 4b^4 = (a^2 -2ab + 2b^2)(a^2 + 2ab + 2b^2) $
Periodic Function:
A function $f$ is said to be periodic, with period $k,$ if $f(x+k)=f(x)$ for all $x$
Pigeon Hole Principle:
If more than $n$ objects are distributed in $n$ boxes, then, at least, one box, has more than one object in it.
Polynomial Equations:
- Every polynomial equation of degree $n(\geq 1)$ has exactly $n$ roots.
- If a polynomial equation with real coefficients has a complex root $p+i q\left(p, q \in R, q \neq 0, i^{2}=-1\right)$ then, it also has a complex root $p-i q$
- If a polynomial equation with rational coefficients has an irrational root $p+\sqrt{q}(p, q$ rational, $q>0, q$ not the square of any rational number), then it also has an irrational root $p-\sqrt{q}$.
- If the rational number $\frac{p}{q}$ (a fraction in its lowest terms so that $p, q$ are integers, prime to each other, $q \neq 0),$ is a root of the equation
$a_{0} x^{n}+a_{1} x^{n-1}+\ldots+a_{n}=0$
where $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ are integers and $a_{n} \neq 0$ then, $p$ is a divisor of $a_{n}$ and $q$ is a divisor of $a_{0}$.
- A number $\alpha$ is a common root of the polynomial equations $f(x)=0$ and $g(x)=0$ if and only if it is a root of $h(x)=0$ where $h(x)$ is the G.C.D of $f(x)$ and $g(x)$.
- A number $\alpha$ is a repeated root of a polynomial equation of $f(x)=0$ if and only if it is a common root of $f^{\prime}(x)=0$ and $f(x)=0$
- If $\alpha, \beta, \gamma$ are the roots of the equation $a x^{3}+b x^{2}+c x+d=0,$ then the following relations hold:
- $\alpha+\beta+\gamma=\frac{-b}{a}$
- $\alpha \beta+\beta \gamma+\gamma \alpha=\frac{c}{a}$
- $\alpha \beta \gamma=\frac{-d}{a}$
- If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $a x^{4}+b x^{3}+c x^{2}+d x+e=0,$ then
- $\alpha+\beta+\gamma+\delta=\frac{-b}{a}$ (i.e $\left.\sigma \alpha=\frac{-b}{a}\right)$
- $\alpha \beta+\alpha \gamma+\alpha \delta+\beta \gamma+\beta \delta+\gamma \delta=\frac{c}{a}$ (i.e. $\left.\sigma \alpha \beta=\frac{c}{a}\right)$
- $\alpha \beta \gamma+\alpha \beta \delta+\alpha \gamma \delta+\beta \gamma \delta=\frac{-d}{a}$
- $\alpha \beta \gamma=\frac{e}{a}$.
- An equation containing (involving) an unknown function is called a functional equation.
