The Fundamental Theorem of Algebra - If f(x) is a polynomial of degree n, where n > 0, f has at least one zero in the complex number system.
The fundamental theorem of algebra also gives you the linear factorization theorem.
Linear Factorization Theorem - If f(x) is a polynomial of degree n where n > 0, f has precisely n linear factors
Where c1, c2, . . . , cn are complex numbers
Real and Complex Zeros of Polynomial Functions
Example:
This polynomial can be factored into:
This shows that the zeros are
(x-2) with a multiplicity of two, (x-2i) and (x+2i)
Notice that (x-2) is listed twice and (x-2i) and (x+2i) are both complex zeros.
Remember:
***The n zeros of a polynomial function can be real or complex and they may be repeated***
***Imaginary roots always come in conjugate pairs***
Finding the Zeros of a Polynomial Function
Example:
The possible rational zeros of this polynomial function are ±1, ±2, ±4, and ±8
(Find these by taking factors of the constant divided by the factors of the leading coefficient. The leading coefficient is 1 so we only need the factors of the constant)
By graphing this function, you can see that -2 and 1 are both real zeros.
***Only the real zeros are shown on the graph as x-intercepts***
After finding the real zeros, you can find the imaginary zeros by using synthetic division.
Using -2 and 1 twice you are left with:
To find the zeros we need to factor this
This gives us the remaining non-real zeros. All five zeros of this polynomial are:
1, 1, -2, 2i, and -2i
Conjugate Pairs
In the last example two of the zeros were -2i and 2i. This is because all imaginary roots must come in conjugate pairs, in the form a + bi and a - bi
In a polynomial function with real coefficients if a + bi is a zero of the function and b is not 0 then the conjugate a - bi must also be a zero of the function.
Finding a Polynomial with Given Zeros
Given the degree of a polynomial function with real coefficients and its zeros we can easily determine the polynomial.
Example:
A third degree polynomial function with real coefficients has 2 and 4i as zeros.
Given 4i as a zero we know that -4i must also be a zero. Because the polynomial function is a third degree function we now have all the zeros: 2, 4i, and -4i.
We can write these out as:
To get the original polynomial we can multiply these factors.
This results in:
Factoring A Polynomial
Factors of a Polynomial - Every Polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros.
Example:
This can be factored into two quadratic polynomials
This can be further factored to include imaginary numbers
Zeros of a Polynomial Function
Given one complex zero of a polynomial function we can find the remaining zeros.
Example:
5i is a zero of
Because 5i is a complex number, we know that -5i must also be a zero of this polynomial because all complex zeros come in conjugate pairs.
We can multiply these together to get part of the polynomial:
We can now use long division using this to find the remaining zero.
/
This gives us
The zeros of this polynomial are:
