Theory of Equations & Polynomials

Foundational Definition

An $n^{\text{th}}$-degree polynomial equation over $\mathbb{C}$ is given by:

$$P(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0=0(a_n\ne 0)$$

By the Fundamental Theorem of Algebra, $P(x)=0$ possesses exactly $n$ complex roots ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n$.

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1. Vieta's Formulas (Relations Between Roots and Coefficients)

📊Visual Mathematical Intuition

Quadratic Parabola Loci, Vertex & Location of Roots

Parabola Vertex:$V\left(-\frac{b}{2a}, -\frac{D}{4a}\right)$

Discriminant Cases:$D > 0 \text{ (2 Real)}, \; D = 0 \text{ (1 Real)}, \; D < 0 \text{ (Complex)}$

Trapping in (k1, k2):$a \cdot f(k_1) > 0, \; a \cdot f(k_2) > 0, \; k_1 < -\frac{b}{2a} < k_2$

Parabolic curve geometry showing vertex $(-b/2a, -D/4a)$, discriminant root conditions ($\Delta > 0, = 0, < 0$), and boundary interval $(k_1, k_2)$ trapping criteria.

Let ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n$ be the roots of $P(x)=0$. Expanding $a_n\prod _{k=1}^n(x-{\alpha }_k)=0$ gives:

$$\begin{array}{rl}{S}_1& =\sum _{i=1}^n{\alpha }_i=-\frac{a_{n-1}}{a_n}\\ S_2& =\sum _{1\le i<j\le n}{\alpha }_i{\alpha }_j=\frac{a_{n-2}}{a_n}\\ S_3& =\sum _{1\le i<j<k\le n}{\alpha }_i{\alpha }_j{\alpha }_k=-\frac{a_{n-3}}{a_n}\\ & ⋮\\ S_n& =\prod _{i=1}^n{\alpha }_i=(-1{)}^n\frac{a_0}{a_n}\end{array}$$

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2. Newton's Sums for Polynomial Roots

Newton-Girard Power Sum Theorem

Let $p_k=\sum _{i=1}^n{\alpha }_i^k={\alpha }_1^k+{\alpha }_2^k+\cdots +{\alpha }_n^k$ be the power sum of degree $k$. For $k\ge n$:

$$a_n{p}_k+a_{n-1}{p}_{k-1}+a_{n-2}{p}_{k-2}+\cdots +a_0{p}_{k-n}=0$$

High-Speed Application (JEE Advanced Pattern): For quadratic $a{x}^2+bx+c=0$ with roots $\alpha ,\beta$ and $p_n=a_n={\alpha }^n+{\beta }^n$:

$$a{p}_n+b{p}_{n-1}+c{p}_{n-2}=0⟹a{p}_n+b{p}_{n-1}=-c{p}_{n-2}$$

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3. Location of Roots for Quadratic Equations

Let $f(x)=a{x}^2+bx+c$ ($a>0$). The discriminant $D=b^2-4ac$.

Condition on Roots	Necessary & Sufficient Conditions
Both roots greater than a real number $k$ ($\alpha ,\beta >k$)	$D\ge 0,-\frac{b}{2a}>k,f(k)>0$
Both roots less than a real number $k$ ($\alpha ,\beta <k$)	$D\ge 0,-\frac{b}{2a}<k,f(k)>0$
A number $k$ lies between the roots ($\alpha <k<\beta$)	$f(k)<0$ (automatically implies $D>0$)
Both roots lie strictly within interval $(k_1,k_2)$	$D\ge 0,k_1<-\frac{b}{2a}<k_2,f(k_1)>0,f(k_2)>0$
Exactly one root lies in interval $(k_1,k_2)$	$f(k_1)\cdot f(k_2)<0$

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4. Condition for Common Roots

One Common Root

If $a_1{x}^2+b_{1}x+c_1=0$ and $a_2{x}^2+b_{2}x+c_2=0$ have a common root $\alpha$:

$$(c_1{a}_2-c_2{a}_1{)}^2=(a_1{b}_2-a_2{b}_1)(b_1{c}_2-b_2{c}_1)$$

Both Roots Common

$$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$$

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5. Descartes' Rule of Signs

Bound on Real Roots

The number of positive real roots of a polynomial $P(x)$ with real coefficients cannot exceed the number of sign variations in the sequence of its coefficients, and differs from it by an even non-negative integer.
The number of negative real roots is bounded similarly by the sign variations in $P(-x)$.