1) Polynomial equations have as many roots as the highest power of the variable. The roots can be real or complex, and may be repeated. 2) Quadratic equations can be solved by setting the coefficients equal to functions of the roots, or by factorizing the quadratic expression. 3) Cubic equations have three roots that relate to the coefficients, and their symmetrical functions can be written in terms of sums and products of the roots.

1. Roots of Polynomial Equations
Basics
• Polynomials have one root for each power of x. Quadratics have 2
roots, cubics 3 roots etc.
• The roots can be identical (repeated roots) and do not have to be real
(if there are complex roots then they come in complex conjugate pairs).
• The polynomial equation can be reconstructed from the roots by
multiplying factors: (x – α)(x – β)… = 0
Quadratic Equations
By co-efficients By factors
ax2 + bx + c = 0 (x – α)(x – β)= 0
2
b c x – (α + β)x + αβ = 0
x2 x 0
a a
b c
Equating co-efficients shows that: ;
a a
Symmetrical Functions
We can write many functions of α and β. Those which are unchanged by
swapping α and β are said to be symmetrical.
2 2 1 1
Examples; ; ; ;
All symmetrical functions can be written in terms of the two basic functions:
α + β; αβ. For Example;
2 2 2
2
1 1
3 3 3
3

2. Creating new equations
We can create equations with roots related to the original equation. There are
two ways to do this (a) by working out the new values of α + β and αβ and (b)
by directly developing the new equation.
Example: If the roots of x2+3x+10 = 0 are α and β, find the equation whose
roots are 3α and 3β
By calculation Directly
α + β = -3; αβ = 10 Let u = 3α
3α + 3β = 3(α + β) = -9; Then α = u/3 but α satisfies x2+3x+10
(3α)(3β) = 9αβ = 90 =0
2
So α + 3α +10 =0
So new equation is u2 3u
x2 + 9x + 90 = 0 10 0
9 3
u2 + 9u + 90 =0
Cubic Equations
The cubic equation ax3 + bx2 + cx + d = 0 has roots α, β and γ. Equating
b c d
co-efficients shows that: ; ;
a a a
Symmetrical functions must be unchanged by the interchange of any two of
α, β and γ. So the following are NOT symmetrical:
2 2 2
; ;
All symmetrical functions can be written in terms of the three basic functions:
; ;
Relationship between roots
If roots in arithmetic progression then use -d, , +d. Hence the sum of roots
is just 3 and thus one root is equal to –b/3a
If the roots are in geometric progression use /r, , r. Hence the product of
3
d
3
roots is equal to and then thus one of the roots is equal to .
a