2x^5-11x^4+69x^3+135x^2-675x

Question

Find the complex zeros of this function using the zero 3-6i and the write the linear factorization of the function

Answer 1

2x⁵-11x⁴+69x³+135x²-675x=

x(2x⁴-11x³+69x²+135x-675).

Since 3-6i is a zero, then so is 3+6i.

If we combine them we get (x-3+6i)(x-3-6i)=x²-6x+9+36=

x²-6x+45 as a factor of the quartic in parentheses.

x²-6x+45 ) 2x⁴-11x³+69x²+135x-675 | 2x²+x-15

                 2x⁴-12x³+ 90x²

                          x³-   21x²+135x

                          x³-   6x²+   45x

                               -15x²+  90x-675

We can find the zeroes of 2x²+x-15=(2x-5)(x+3). So the remaining zeroes are 5/2 and -3.

The zeroes are therefore 0, 3-6i, 3+6i, 5/2, -3. The only complex factors are 3-6i and 3+6i.

Factored into real factors we have x(2x-5)(x+3)(x²-6x+45).