If a polynomial function f(x) has roots -8, 1, and 6i, what must also be a root of f(x)?
Solution:
Every complex number has another complex number associated with it, known as the complex conjugate.
A complex conjugate of a complex number is another complex number that has the same real part as the original complex number and the imaginary part has the same magnitude but opposite sign.
The product of a complex number and its complex conjugate is a real number.
Complex roots are in pairs that are both positive and negative.
If 6i is a root then - 6i is also a root
We know that
f (x) = (x + 8) (x - 1) (x2 + 36)
Now equate each factor to zero
x + 8 = 0
x = - 8
x - 1 = 0
x = 1
x2 + 36 = 0
x2 = - 36
x = √-36
So we get
x = ±6i
Therefore, the root of f (x) is ±6i.
If a polynomial function f(x) has roots -8, 1, and 6i, what must also be a root of f(x)?
Summary:
If a polynomial function f(x) has roots -8, 1, and 6i, - 6i is also a root of f(x).
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