Answer to Question #116857 in Algebra for jay

Question

) Given that 3 + i is a root of the equation z − 3z − 8z + 30 = 0, find the
remaining roots.

Answer 1

Solution: According to the complex conjugate root theorem: if 3+i is a root, then 3-i is a root too. In this way the original polynomial P=z³-3z²-8z+30=(z-z₁)(z-3-i)(z-3+i)=(z-z₁)((z-3)²-i²)=(z-z₁)(z²-6z+10);To find remaing roots - divide the original polynomial by z²-6z+10:z³-3z²-8z+30=z*(z²-6z+10)+3*(z²-6z+10)=(z+3)(z²-6z+10);I.e. the remaining root is equal to -3.
Answer: 3-i, -3.