find a polynomial F(x) of degree 3 with real coeffiencts and the following zeroes. -1, -1+3i

Question

campbell_st · Answer

ok... so if looking at the zeros

x = -1 is a zero then x + 1 is a factor

the complex zero x = -1 + 3i as a conjugate pair   x = -1 - 3i

this means that you are looking at a quadratic factor 

so to find the quadratic start with 

$$x = -1 \pm 3i$$

so add 1 to both sides 

$$x + 1 = \pm 3i $$

now square both sides

$$(x + 1)^2 = (\pm3i)^2$$

when you square a negative you gte a positive so it can be written as 

$$(x + 1)^2 = 9i^2$$

now you should know 

$$i^2 = -1$$

so you have 

$$(x + 1)^2 = -9$$

or 

$$(x + 1)^2 + 9 = 0$$

expand the left hand side and then collect like terms to get the quadratic factor... 

multiply it by the linear factor shown earlier and you'll have the polynomial. 

hope it helps