If a is a zero of f(x)=0, then (x-a) is a factor of f(x).
If f(x) is a polynomial with real coefficients, then the complex roots of f(x)=0 occur in conjugate pairs.
If f(x) is a polynomial with real coefficients, then the complex roots of f(x)=0 occur in conjugate pairs.
This theorem states that if a + bi is a complex root, then a- bi is also a zero. Additionally, recall that if a is a zero of f(x)=0, then (x-a) is a factor of f(x).
Root
Factor
-3
x-(-3)
1
x-1
-3i
x-(-3i)
3i
x-3i
Polynomial
f(x)= [x-(-3)] (x-1) [x-(-3i)](x-3i)
Let's simplify the polynomial by applying the Distributive Property. For simplicity, we will start by multiplying the first two factors and the last two factors separately.
[x-(-3)] * (x-1)
[x-(-3i)] * (x-3i)
After we find these products, we will multiply the obtained expressions.
