The Chemistry Maths Book, Second Edition

48 Chapter 2Algebraic functions

once. In general it has an odd number of real roots. A polynomial of even degree

(n 1 = 1 2, 4, 6, =) has an even number of real roots, or no real roots if the curve does not

cross the x-axis.

EXAMPLE 2.22Factorization of a cubic

A polynomial of degree 3 can have all three roots real or it can have one real root and

two complex roots; for example, in Figure 2.11,

(a) three real roots: x

3

1 − 16 x

2

1 + 111 x 1 − 161 = 1 (x 1 − 1 1)(x 1 − 1 2)(x 1 − 1 3)

(b) three real roots, one double: x

3

1 − 15 x

2

1 + 17 x 1 − 131 = 1 (x 1 − 1 1)

2

(x 1 − 1 3)

(c) one real root and two complex roots: x

3

1 − 13 x

2

1 + 14 x 1 − 121 = 1 (x 1 − 1 1)(x

2

1 − 12 x 1 + 1 2)

The roots of the quadratic factor are 11 ± 1 i, where

x

2

1 − 12 x 1 + 121 = 1 [x 1 − 1 (1 1 + 1 i)][x 1 − 1 (1 1 − 1 i)]

and the fully factorized form of the cubic is

x

3

1 − 13 x

2

1 + 14 x 1 − 121 = 1 (x 1 − 1 1)(x 1 − 111 − 1 i)(x 1 − 111 + 1 i)

EXAMPLE 2.23Given that x 1 − 11 is a factor, find the roots of the cubic

x

3

1 − 17 x

2

1 + 116 x 1 − 110.

Ifx 1 − 11 is a factor then the cubic function can be written as

x

3

1 − 17 x

2

1 + 116 x 1 − 1101 = 1 (x 1 − 1 1)(ax

2

1 + 1 bx 1 + 1 c)

= 1 ax

3

1 + 1 (b 1 − 1 a)x

2

1 + 1 (c 1 − 1 b)x 1 − 1 c

For this equation to true for all values of xit is necessary that the coefficient of each

power of xbe the same on both sides of the equal sign:

11 = 1 a, − 71 = 1 b 1 − 1 a,16 1 = 1 c 1 − 1 b, − 101 = 1 −c

i=−1,

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Figure 2.11