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
0
x-0
-5
x-(-5)
3+i
x-(3+i)
3-i
x-(3-i)
Polynomial
f(x)= (x-0) [x-(-5)] [x-(3+i)][x-(3-i)]
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-0) & * [x-(-5)]
[x-(3+i)] & * [x-(3-i)]
After we find these products, we will multiply the obtained expressions.
