Theory of Equations:
Eg1. How may roots are there for
f ( x)  ( x  1)( x  3)( x  5)  k 2 ( x  2)( x  4)( x  6)  0
, and where are
they ?
Reciprocal Equations:
Eg2. Solve 6 x 6  25 x 5  31x 4  31x 2  25 x  6  0  f ( x) .
Transformation of Equations:
Eg3. (a) Show ( a + b ) is a root of x 3  3abx  (a 3  b 3 )  0 .
(b) (i) Express x 3  6 x  6  0 … (*) in the above form.
Hence find a real root of (*).
(ii) By the transformation x  y   , convert
x 3  3x 2  3x  11  0 …(**)
in the form of y 3  py  q  0 .
Hence find a root of (**).
Relation between roots and coefficients
Eg4. Find a necessary and sufficient condition for the sum of 2 roots of
x 4  px 3  qx 2  rx  s  0 to be equal to the sum of the other two roots.
Eg5. If 1,  1 ,  2 ,........,  2 n are the roots of x 2 n 1  1  0 , find a polynomial
equation whose roots are 0, (1  1 ), (1   2 ),........, (1   2n ) .
Hence find (1  1 )(1   2 )......(1   2 n ) .
Multiple Roots:
Eg6. (a) Prove that f (x) has α as multiple root iff f ( )  f ' ( )  0 .
(b) Prove x 3  px  q  0 has a repeated root iff 4 p 3  27q 2  0 .
Reduce x 3  x 2  8 x  12  0 to the form x 3  px  q  0 and hence, or
otherwise, solve it.