Search 75,804 tutors
FIND TUTORS
Ask a question
0 0

show that -1 is a zero multiplicity 3 of P(x)=X^5+9X^4+33x^3+55X^2+42x+12, find all zeros (real and complex) for P(x), write P(x) in factored form and graph

Tutors, please sign in to answer this question.

1 Answer

If -1 is a zero of multiplicity 3 of a polynomial then (x-(-1))=x+1 divides P(x) three times, or (x+1)³divides P(x) without remainder. The limitations of notation force me to show that using synthetic division three times we get
 
-1       1   9    33   55   42   12
             -1   -18  -25  -30  -12
-1       1   8    25   30   12
              -1   -7   -18  -12
-1       1   7   18   12
              -1  -6    -12
          1   6   12
The answer is x²+6x+12, whose roots are -3±i√3 by the quadratic formula.  I wish I knew how to write out the long division.
So, finally, we have that
P(x)=(x-(-1))³(x-(-3+i√3))(x-(-3-i√3))=(x+1)³(x+3-i√3)(x+3-i√3), the factored form