If a is a root of P(x)=0, then (x-a) is a factor of P(x). If P(x) is a polynomial with real coefficients, then the complex roots of P(x)=0 occur in conjugate pairs.
If P(x) is a polynomial with real coefficients, then the complex roots of P(x)=0 occur in conjugate pairs.
This theorem states that if a + bi is a complex root, then a - bi is also a root. Additionally, recall that if a is a root of P(x)=0, then (x-a) is a factor of P(x).
Root
Factor
3+i
x-(3+i)
3-i
x-(3-i)
2
x-2
-4
x-(-4)
Polynomial
P(x)= (x-(3+i)) (x-(3-i)) (x-2) (x-(-4))
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+i)) * (x-(3-i))
(x-2) * (x-(-4))
After we find these products, we will multiply the obtained expressions.
