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56 CHAPTER 3. POLYNOMIAL EQUATIONS
However, if b 2 − 4ac = 0, i.e. c = b 2 /4a then there is only one solution: x = −b
In this case, the equation becomes ax 2 + bx + b2
4a
= 0, which can be re-written
as a ( x +
2a) b 2
= 0, making it more obvious that there is a repeated root, and
that the polynomial is not square-free (definition 32).
Mathematicians dislike the sort of anomaly in “this equations has two solutions
except when c = b 2 /4a”, especially as there are two roots as c tends to
the value b 2 /4a. We therefore say that, in this special case, x = −b
2a
is a double
root of the equation. This can be generalised, and made more formal.
2a .
Definition 33 If, in the equation f = 0, f has a square-free decomposition
f = ∏ n
i=1 f i i, and x = α is a root of f i, we say that x = α is a root of f of
multiplicity i. When we say we are counting the roots of f with multiplicity,
we mean that x = α should be counted i times.
Proposition 17 The number of roots of a polynomial equation over the complex
numbers, counted with multiplicity, is equal to the degree of the polynomial.
Proof. deg(f) = ∑ ideg(f i ), and each root of f i is to be counted i times as a
root of f. That f i has i roots is the so-called Fundamental Theorem of Algebra.
In this case, the two roots are given by the two possible signs of the square
root, and √ 0 is assumed to have both positive and negative signs.
3.1.2 Cubic Equations
There is a formula for the solutions of the cubic equation
x 3 + ax 2 + bx + c, (3.6)
albeit less well-known to schoolchildren:
√
1 3
36 ba − 108 c − 8 a
6
3 + 12 √ 12 b 3 − 3 b 2 a 2 − 54 bac + 81 c 2 + 12 ca 3 −
2b − 2 3
√ a2
36 ba − 108 c − 8 a3 + 12 √ 12 b 3 − 3 b 2 a 2 − 54 bac + 81 c 2 + 12 ca − 1 3 3 a.
3
We can simplify this by making a transformation 1 to equation (3.6): replacing
x by x − a 3
. This transforms it into an equation
x 3 + bx + c (3.7)
(where b and c have changed). This has solutions of the form
√
1 3
−108 c + 12 √ 12 b
6
3 + 81 c 2 −
2b
√ √
−108 c + 12 12 b3 + 81 c . (3.8)
2
1 This is the simplest case of the Tschirnhaus transformation[vT83], which can always
eliminate the x n−1 term in a polynomial of degree n.
3