TN 12th Maths — Learn, Visualize, Practice

1. Polynomials, roots and multiplicity

A polynomial equation of degree $n$ can have at most $n$ roots. Counting each root as often as it repeats (its multiplicity), it has exactly $n$ roots in the complex numbers.

DefinitionFundamental Theorem of Algebra

Every polynomial equation $P(x) = 0$ of degree $n \ge 1$ has at least one root in $\mathbb{C}$ . Repeatedly factoring, it splits into exactly $n$ linear factors over $\mathbb{C}$ .

Note: $a$ is a root of multiplicity $k$ if $(x-a)^k$ divides $P(x)$ but $(x-a)^{k+1}$ does not. For example, $x^2 - 6x + 9 = (x-3)^2$ has $3$ as a root of multiplicity $2$.

2. Vieta's formulae — roots ↔ coefficients

The roots of a polynomial are locked to its coefficients. You never need to solve the equation to know the sum or product of its roots — Vieta's formulae read them straight off.

Quadratic ax² + bx + c = 0, roots α, β

\alpha+\beta = -\dfrac{b}{a}, \qquad \alpha\beta = \dfrac{c}{a}

Cubic ax³ + bx² + cx + d = 0, roots α, β, γ

\alpha+\beta+\gamma = -\dfrac{b}{a},\quad \alpha\beta+\beta\gamma+\gamma\alpha = \dfrac{c}{a},\quad \alpha\beta\gamma = -\dfrac{d}{a}

Key idea: Drag the coefficients in the Visualize tab and watch these three sums update alongside the graph's roots.

Example

The roots of

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

are

\alpha, \beta, \gamma

. Find

\alpha+\beta+\gamma

and

\alpha\beta\gamma

.

3. Forming an equation from its roots

Vieta's formulae run in reverse: given the roots, you can write down the polynomial. For a quadratic with roots $\alpha, \beta$ :

x^2 - (\alpha+\beta)\,x + \alpha\beta = 0

Example

Form the quadratic whose roots are

3

and

-2

.

4. Complex Conjugate Root Theorem

DefinitionConjugate roots

If a polynomial has real coefficients and $a + bi$ ( $b \ne 0$ ) is a root, then its conjugate $a - bi$ is also a root. Non-real roots always come in pairs.

Note: A consequence: a polynomial of odd degree with real coefficients must have at least one real root, because complex roots pair up and cannot account for an odd count.

5. Rational Root Theorem

For hunting exact rational roots, this theorem gives a short finite list of candidates to test.

DefinitionRational Root Theorem

For a polynomial with integer coefficients, any rational root $\dfrac{p}{q}$ in lowest terms has $p$ dividing the constant term $a_0$ and $q$ dividing the leading coefficient $a_n$ .

Example

List the candidate rational roots of

2x^3 + x - 6 = 0

.

6. Descartes' Rule of Signs

Before solving, you can bound how many positive and negative real roots are even possible, just by counting sign changes.

The number of positive real roots equals the sign changes in $P(x)$ , or is fewer by an even number.
The number of negative real roots is found the same way from $P(-x)$ .

Example

What does Descartes' rule say about

P(x) = x^3 - 2x^2 + x - 5

?

7. Reciprocal & special equations

If the coefficients sum to zero, then $x = 1$ is a root (since $P(1) =$ sum of coefficients).
Equations in only even powers reduce to a polynomial in $x^2$ — solve, then take square roots.
Reciprocal equations have roots in pairs $r$ and $\tfrac{1}{r}$ ; their coefficients read the same forwards and backwards.