The Chemistry Maths Book, Second Edition

2.5 Polynomials 49

Thena 1 = 11 ,b 1 = 1 − 6 ,c 1 = 110 , and the quadratic factor is

x

2

1 − 16 x 1 + 1 10 with roots

(see also Examples 2.25 and 2.26). The cubic is therefore of

type (c) in Example 2.22, with one real root,x

1

1 = 12 and the

two complex rootsx

2

1 = 131 + 1 iandx

3

1 = 131 − 1 i. The graph of the

function, Figure 2.12, shows however that, like types (a) and

(b), the function has local maximum and minimum values

(or turning points; see Section 4.10) at x 1 = 12 and x 1 = 1823 ,

respectively.

0 Exercises 53–55

EXAMPLE 2.24Factorization of a quartic

Three cases may be considered.

(i) 4 real roots; for example

x

4

1 − 1 x

3

1 − 17 x

2

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

(ii) 2 real roots and 2 complex roots; for example

x

4

1 − 12 x

3

1 + 1 x

2

1 + 12 x 1 − 121 = 1 (x 1 − 1 1)(x 1 + 1 1)(x

2

1 − 12 x 1 + 1 2)

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

(iii) 4 complex roots; for example

x

4

1 − 12 x

3

1 + 13 x

2

1 − 12 x 1 + 121 = 1 (x

2

1 + 1 1)(x

2

1 − 12 x 1 + 1 2)

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

0 Exercise 56

Examples 2.22 to 2.24 demonstrate that, if complex numbers are disallowed, a polynomial

can always be factorized as the product of some linear factors, one for each real root,

and, at most, quadratic factors, all real.

6

The theorem is used in Section 2.7 for the

construction of partial fractions.

xi=

±−

=±

63640

2

3

y

x

1

2

8 / 3

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Figure 2.12

6

This is the statement of the fundamental theorem of algebra given in Gauss’ first proof of 1799.