SOLUTION: Show that y+1 is a factor of the expression y^3+(m-2)y^2+(m-7)y-4 for all values of m. If the expression also has a factor y+2, find the value of m and the third factor.

Question 695486: Show that y+1 is a factor of the expression y^3+(m-2)y^2+(m-7)y-4 for all values of m. If the expression also has a factor y+2, find the value of m and the third factor.
Answer by mouk(232) (Show Source): You can put this solution on YOUR website!
Let
By the factor theorem is a factor of if and only if
So is a if and only if , Now

so showing is a factor QED

Similarly, so,

giving
So substituting , gives:

We now know that both and are factors
Suppose the last factor is then

Multiplying out gives:

So Comparing with the constant coefficient of
gives: so
Hence the third factor is

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