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.

Algebra ->  Test  -> Lessons -> 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.      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
Let +f%28y%29+=+y%5E3%2B%28m-2%29y%5E2%2B%28m-7%29y-4+
By the factor theorem +y-A+ is a factor of +f%28y%29+ if and only if +f%28A%29=0+
So +y%2B1+ is a if and only if +f%28-1%29+=+0+, Now
+f%28-1%29+=+%28-1%29%5E3%2B%28m-2%29%28-1%29%5E2%2B%28m-7%29%28-1%29-4+
+f%28-1%29+=++-1+%2B+%28m-2%29+-+%28m-7%29+-+4+
+f%28-1%29+=++0+
so +f%28-1%29+=+0+ showing +y%2B1+ is a factor QED

Similarly, +f%28-2%29=0+ so,
+%28-2%29%5E3+%2B+%28m-2%29%28-2%29%5E2+%2B+%28m-7%29%28-2%29+-+4+=+0+
+-8+%2B+4%28m-2%29+-2%28m-7%29+-+4+=+0+
+-8+%2B+4m-8+-2m+%2B14+-+4+=+0+
+2m+-+6+=0++ giving +m+=+3+
So substituting +m+=+3+, gives:
+f%28y%29+=+y%5E3+%2B+%283-2%29y%5E2+%2B+%283-7%29y+-+4+
+f%28y%29+=+y%5E3+%2B+y%5E2+-+4y+-+4+
We now know that both +y+%2B+1+ and +y+%2B+2+ are factors
Suppose the last factor is +y%2Bn+ then
+f%28y%29+=+y%5E3+%2B+y%5E2+-+4y+-+4+
+f%28y%29+=+%28y%2B1%29%28y%2B2%29%28y%2Bn%29+
Multiplying out gives:
+f%28y%29+=+%28y%5E2+%2B+3y+%2B+2%29%28y%2Bn%29+
+f%28y%29+=+y%5E3+%2B+3y%5E2+%2B+2y+%2B+ny%5E2+%2B+3ny+%2B+2n+
+f%28y%29+=+y%5E3+%2B+%283%2Bn%29y%5E2+%2B+%283n%2B2%29y+%2B+2n+
So Comparing with the constant coefficient of +f%28y%29+=+y%5E3+%2B+y%5E2+-+4y+-+4+
gives: +2n+=+-4+ so +n=-2+
Hence the third factor is +y-2+