The Chemistry Maths Book, Second Edition

2.5 Polynomials 45

EXAMPLE 2.17Case

The quadratic

2 x

2

1 − 18 x 1 + 181 = 1 2(x 1 − 1 2)

2

has zero discriminant and the double root(two equal roots)x 1 = 1 2.

0 Exercises 47, 48

When the discriminant is negative, formula (2.20) requires the taking of the square

root of a negative number, and the result is not a real number. In this case the roots of

the quadratic are a pair of complex numbers involving the square root of

(see Section 8.2)

EXAMPLE 2.18Case

The quadratic

x

2

1 − 14 x 1 + 113

has a pair of complex roots x

1

and x

2

given by

The roots arex

1

1 = 121 + 13 iandx

2

1 = 121 − 13 iand the factorized form of the quadratic is

(x 1 − 1 x

1

)(x 1 − 1 x

2

).

0 Exercises 49, 50

0 Exercises 51, 52

For very large (‘large enough’) values of |x|the term in x

2

in the quadratic

f(x) 1 = 1 ax

2

1 + 1 bx 1 + 1 cis very much larger in magnitude than the other two terms. Thus,

dividing the function byx

2

,

This means that for large enough positive and negative values of x, the function behaves

like the simpler function ax

2

, and can sometimes be replaced by it. In general for the

polynomial of degree n, equation (2.9),

fx

x

a

b

x

c

x

ax

()

22

=+ + → as →±∞

xi=

±−

=±

436

2

23

bac

2

−< 40.

−=−11:i

bac

2

−= 40.