Given that 3i is a zero of q , write the polynomial in factored form as a product of linear factors

Answer 1

First you need to do long division of q divided by (x² + 9) since it's given that 3i is a zero and assuming that -3i is also a zero. Then you get x⁴ + 5x³ - 4x² - 50x - 60. Guess and check to see that -2 is a zero so divide x⁴ + 5x³ - 4x² - 50x - 60 by x + 2 and get x³ + 3x² - 10x - 30

Factoring by grouping:

x³ + 3x² - 10x - 30

= x² ( x + 3 ) - 10 ( x + 3 )

= ( x² - 10 ) ( x + 3 )

= ( x + √10 ) ( x - √10 ) ( x + 3 )

And putting all the factors together,

q(x) = (x+3i) (x-3i) (x+2) (x+√10) (x-√10) (x+3)