Theory of Equations: Mathematics Notes for IIT-JEE (Main + Advanced)

Theory of Equations (IIT‑JEE Mathematics)

Goal – Understand how the coefficients of a polynomial determine its roots and how to manipulate equations to find or use those roots.

A polynomial equation is an equation of the form

[ a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0=0,\qquad a_n\neq 0 ]

Its degree is (n).
The roots (or zeros) of the equation are the values of (x) that satisfy it.
Theory of equations studies the relationship between the coefficients ((a_i)) and the roots ((\alpha_i)).
Key tools:
- Vieta’s formulas – express symmetric sums of the roots in terms of coefficients.
- Discriminant – tells whether roots are real/complex and distinct/repeated.
- Rational Root Test – helps locate rational roots.
- Factorisation – reduces a high‑degree polynomial to lower‑degree factors using the roots we find.

Concept	What it is	Formula / Idea
Polynomial	Sum of monomials (a_kx^k).	(p(x)=\sum_{k=0}^n a_k x^k)
Degree	Highest power with non‑zero coefficient.	(\deg p(x)=n)
Root	(x=r) such that (p(r)=0).	(x-r) is a factor
Multiplicity	Number of times a root occurs.	If ((x-r)^m) divides (p(x)) but ((x-r)^{m+1}) does not
Vieta’s formulas	Symmetric sums of roots in terms of coefficients.	For monic polynomial (x^n + a_{n-1}x^{n-1}+\dots+a_0=0)
[
\sum_i \alpha_i = -a_{n-1},;
\sum_{i<j} \alpha_i\alpha_j = a_{n-2},;
\dots,;
\prod_i \alpha_i = (-1)^n a_0
]
Discriminant	Determines nature of roots (real/complex, distinct/repeated).	For quadratic: (\Delta=b^2-4ac); for cubic: (\Delta=18abcd-4b^3d+b^2c^2-4ac^3-27a^2d^2)
Rational Root Test	Any rational root (p/q) (in lowest terms) satisfies (p \mid a_0) and (q \mid a_n).	For monic polynomial, candidate roots are divisors of (a_0).
Factorisation	Split into linear and irreducible quadratic factors over (\mathbb R).	(p(x)=a_n\prod (x-\alpha_i)) or (\prod (x^2+bx+c)).
Symmetric polynomials	Polynomials invariant under any permutation of the roots.	Expressible via elementary symmetric sums (Vieta).
Complex Conjugate Root Theorem	Non‑real roots of polynomials with real coefficients occur in conjugate pairs.	If (\alpha) is a root, so is (\overline{\alpha}).

Find all real roots of

[ x^3-6x^2+11x-6=0 ]

and verify them using Vieta’s formulas.

Observe: The polynomial is monic, degree 3.
Vieta’s formulas for a monic cubic (x^3+px^2+qx+r=0):

[ \begin{aligned} \alpha+\beta+\gamma &= -p\ \alpha\beta+\beta\gamma+\gamma\alpha &= q\ \alpha\beta\gamma &= -r \end{aligned} ]

Here (p=-6,; q=11,; r=-6).
Try rational roots (Rational Root Test).
Candidates: (\pm1,\pm2,\pm3,\pm6).
Test (x=1): (1-6+11-6=0).
So (x=1) is a root.
Divide by ((x-1)) (synthetic division):

[ x^3-6x^2+11x-6 = (x-1)(x^2-5x+6) ]
Solve the quadratic (x^2-5x+6=0):

[ x=\frac{5\pm\sqrt{25-24}}{2}=\frac{5\pm1}{2}\Rightarrow x=3,;x=2 ]
Roots found: (\boxed{1,;2,;3}).
Check with Vieta:
- Sum: (1+2+