Download Roots of Polynomial Equations

Roots of polynomial equations
Fundamental Theorem of Algebra
Fundamental Theorem Of Algebra :
For all non-constant polynomials with ai  ,
an z n  an 1 z n 1 
 a1 z  a0  an ( z  1 )( z   2 )
( z   n 1 )( z   n ),
where  i 
Roots of a quadratic equation
If  and  are the roots of a quadratic equation, f ( x)  ax 2  bx  c  0, then the equation
must be of the form
f ( x)  k ( x   )( x   ), k constant
k ( x   )( x   )  ax 2  bx  c
So


 k x 2  (   ) x    ax 2  bx  c
Equating coefficients yields
k a
 k (   )  b
k  c
Therefore we obtain
b
a
c
a
     ,   .
We can now think of the quadratic equation as
x 2   sum of roots x   product of roots  0.
Find similar relations for cubics and quartics.
Quadratic, cubic, quartic, quintic
Question
If 2 x 3  x 2  x  1  0 has roots  ,  and  , what is:
(a)      ?
(b)  2   2   2 ?
(c)  3   3   3 ?
1 1 1
(d)   ?

(e)
(f)

1

1

3
2



1

1

3
2


1

1

3
2
?
?
Another question
If x3  x 2  x  1  0 has roots  ,  and  , what is:
(a)      ?
(b)  2   2   2 ?
(c)  3   3   3 ?
1 1 1
(d)   ?

(e)
(f)

1

1

3
2



1

1

Why is this much
easier?
3
2


1

1

3
2
?
?
Identities for roots of a cubic
If the cubic polynomial ax3  bx 2  cx  d has roots  , and 
then
             2       
2
2
2
2
 3   3   3         2   2   2         3


              3         3
2
Do Exercise 8A, p.151
Related roots- examples
Example 1: The cubic equation x3  2 x 2  3x  4  0 has roots  ,  and  .
Find the equation which has roots  2 ,  2 and  2 .
Example 2 : The cubic equation x3  3x 2  4 x  5  0 has roots  ,  and  .
Find the equation which has roots    ,    and    .
Example 3 : The cubic equation x3  9 x 2  31x  39  0 has roots  ,  and 
which are in arithmetic progession. Solve the equation.
Do Exercise 8B, p.153