Class 10 · Chapter 2

Polynomials

Zeroes of a polynomial and their relationship with coefficients; division algorithm for polynomials.

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Detailed theory

\alpha, \beta

are zeroes of

ax^2 + bx + c

(

a\ne 0

), then

\alpha + \beta = -b/a

and

\alpha\beta = c/a

. This lets you construct a quadratic with given zeroes:

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

Key concepts & formulas

For

ax^2+bx+c

: sum of zeroes

=-b/a

, product

=c/a

A quadratic has at most

2

zeroes; a cubic at most

3

Division algorithm:

p(x) = g(x)q(x) + r(x)

\deg r<\deg g

(x-a)

divides

p(x)

iff

p(a)=0

(Factor theorem).

Geometrically, zeroes are

x

-intercepts of

y=p(x)

For a cubic

ax^3+bx^2+cx+d

\alpha+\beta+\gamma=-b/a

\alpha\beta+\beta\gamma+\gamma\alpha=c/a

\alpha\beta\gamma=-d/a