Use the given zero to write p(x) in complete factored form

Question

Use given zero to write p(x) in complete factored form

p(x)=x⁴+x³+9x²+9x  zero: 3i

Answer 1

If once zero is 3i, then the other zero (conjugate) is -3i.  So we can get two factors.

 

(x - 3i) and (x + 3i)

 

 

Multiply these factors to obtain

 

x² + 9

 

 

Now we divide p(x) by (x² + 9) using long division.

 

           

                         x² +   x

             ____________________

(x² + 9) |  x⁴ +   x³ + 9x² + 9x

                x⁴ + 0x³ + 9x²

               ________________

                         x³ +         9x

                         x³ +         9x

                        ________________

                                          0

 

 

So now your factored form of p(x) is

 

p(x) = (x² + 9)(x² + x)

 

 

Then, you factor out the last factor completely by factoring out an x.

 

p(x) = (x² + 9)x(x + 1)

 

p(x) = x(x² + 9)(x + 1)