Chapter 2 Polynomials

Notes

1. Polynomial

A polynomial in one variable x is an expression of the form:

anxn+an−1xn−1+⋯+a1x+a0a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0an​xn+an−1​xn−1+⋯+a1​x+a0​

where n is a non‑negative integer and an≠0a_n \neq 0an​=0.

2. Degree of a Polynomial

The highest power of the variable in a polynomial is called its degree.

Examples:

3x+13x + 13x+1 → Degree 1

x2−4x+5x^2 - 4x + 5x2−4x+5 → Degree 2

x3+2x2+xx^3 + 2x^2 + xx3+2x2+x → Degree 3

3. Zeros of a Polynomial

A real number k is called a zero of a polynomial p(x) if:

p(k)=0p(k) = 0p(k)=0

Zeros can be found by:

Factorization method

Division method

Graphical method

4. Relationship Between Zeros and Coefficients

For a quadratic polynomial:

ax2+bx+cax^2 + bx + cax2+bx+c

If α\alphaα and β\betaβ are its zeros, then:

α+β=−ba\alpha + \beta = \frac{-b}{a}α+β=a−b​

α×β=ca\alpha \times \beta = \frac{c}{a}α×β=ac​

This relation is very important for JAC board exams.

5. Construction of Polynomial

If zeros are given, a polynomial can be formed using:

(x−α)(x−β)(x - \alpha)(x - \beta)(x−α)(x−β)

Multiply and simplify to get the required polynomial.