Question

Find all roots of the equation

6y5 - 13y4 - 6y3 + 17y2 - 4 = 0

Answer 1

help

Answer 2

first you have to simplify to do anything else

Answer 3

simplify and work it out take your time

Answer 4

put y=1,
6*1^5-13*1^4-6*1^3+17*1^2-4=0
23-23=0
0=0
which is true.
Hence y=1 is a root.
y-1 is a factor of this eq.
by synthetic division,

(y-1)(6y^4-7y^3-13y^2+4y+4)=0
\[6y ^{4}-7y ^{3}-13y ^{2}+4y+4=0\]
y+1=0,gives y=-1
\[put y=2 \in6y ^{3}-13y ^{2}+4=0\]
\[6*2^{3}-13*2^{2}+4=0,52-52=0,0=0\]
Hence y=2 is a root.

\[6y ^{2}-y-2=0\]
\[6y ^{2}-4y+3y-2=0\]
2y(3y-2)+1(3y-2)=0
\[\left( 3y-2 \right)\left( 2y+1 \right)=0,y=\frac{ 2 }{ 3 },\frac{ -1 }{ 2 }\]
roots are 1,1,2,2/3,-1/2