SOLUTION: Factor the Polynomial completely p(x)= x^5+6x^3+9x Find all its zeros. State the multiplicity of each zero. Im specifically not understanding how to fund multiplicity, so

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor the Polynomial completely p(x)= x^5+6x^3+9x Find all its zeros. State the multiplicity of each zero. Im specifically not understanding how to fund multiplicity, so       Log On


   

Question 1099282: Factor the Polynomial completely
p(x)= x^5+6x^3+9x
Find all its zeros. State the multiplicity of each zero.
Im specifically not understanding how to fund multiplicity, so please dont forget that part.
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.
Factor the Polynomial completely
p(x)= x^5+6x^3+9x
~~~~~~~~~~~~~~~~~~~~ 

x%5E5%2B6x%5E3%2B9x = x%2A%28x%5E4+%2B+6x%5E2+%2B+9%29 =    


     ! Notice that x^4 + 6x^2 + 9 = %28x%5E2%29%5E2+%2B+2%2A3%2Ax%5E2+%2B+9 = %28x%5E2%2B3%29%5E2.  
       Therefore, you can continue this chain of equalities in this way


= x%2A%28x%5E2%2B3%29%5E2 =


      The only real zero is z= 0.  All other zeroes are complex zeroes.
      So, you can continue in the complex domain


= x%2A%28x-3i%29%5E2%2A%28x%2B3i%29%5E2.


      Now you can see that there is one real zero x= 0 of multiplicity 1,

                                        complex zero x= 3i of multiplicity 2, and

                                        complex zero  x= -3i of multiplicity 2.