Problema Solution
Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value
n=3
-4 and 1+4i are zeros
f(1)=80
Answer provided by our tutors
We will use the Complex Conjugate Theorem that states:
If P is a polynomial in one variable with real coefficients, and 'a + bi' is a root of P with a and b real numbers, then its complex conjugate 'a − bi' is also a root of P.
So, if 1+4i is one of the zeros then 1-4i is also one of the zeros.
The polynomial can be written as:
f(x) = a(x - (-4))(x - (1 + 4i))(x - (1 - 4i)), where 'a' is constant
........
click here to see the step by step simplification
........
f(x) = ax^3 + 2ax^2 + 9ax + 68a
Next we will find 'a 'by using f(1) = 80
a*1^3 + 2a*1^2 + 9a*1 + 68a = 80
........
click here to see the equation solved for a
........
a=1
This means the 3-degree polynomial with zeros -4 and 1+4i is:
f(x) = x^3 + 2x^2 + 9x + 68