write 4th degree polynomial with zeros at 3,-2, and 3i

Question

im looking for the polynomial with zeros at 3,-2, and 3i. step by step i would like to learn

 

Answer 1

Your roots are

 

x₁ = 3

x₂ = -2

x₃ = 3i

x₄ = -3i

 

x₃ and x₄ are conjugates because these roots can be found using the quadratic formula.  Recall that the discriminate of the quadratic formula is the square-root part.  The value under the square-root is   b² - 4ac.

 

When b² < 4ac, we will have a negative value under the square-root, indicating an imaginary number i. When the values of a,b, and c are plugged in, we have complex roots

 

x = 0 ± 3i

 

 

Your 4th degree polynomial is

 

f(x) = (x - 3)(x + 2)(x + 3i)(x - 3i)

 

Notice that if we multiply the first terms of each factor, we get x⁴ as the first term of the polynomial.  The value of the exponent of the leading term is 4. Therefore, we have a degree of 4.