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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.