a.) Find all zeros of P(x)=x5+9x3 of P, real and complex
b.) Factor P completely.
a.) We first factor P as follows.
P(x)=x5+9x3Given=x3(x2+9)Factor out x3
We find the zeros of P by setting each factor equal to :
Setting x3=0, we see that x=0 is a zero. More over, setting x2+9=0, we get x2=−9, so x=±3i. So the zeros of P are 0,3i and −3i.
b.) Since the zeros are 0,3i and −3i, by the complex Factorization Theorem P factors as
P(x)=x(x−3i)[x−(−3i)]=x(x−3i)(x+3i)
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