1. 5.7 Apply the Fundamental Theorem of Algebra

2. Here is the graph of f(x). How many real zero’s do you see
Here is the graph of f(x). How many real zero’s do you see? What are their values?
This third degree equation has 3 zeros:
x = 3, x = -2, and x = -3

3. Can be factored as
Here is the graph of f(x).
How many times is f(x)=0?
Since (x-5) is a factor twice (notice the exponent), we say x = 5 is a repeated solution.
We say this third degree equation also has 3 zeros: x = -4, and x = 5 and 5.

4. The equation:
Has three solutions: x = -3 , x = 4i, and x = –4i.
And thus 3 factors: (x+3)(x-4i)(x+4i)
Has four zeros: -2, -2, -2 and 0
And thus four factors: (x+2)(x+2)(x+2)(x) or

5. Fundamental Theorem of Algebra
The following important theorem, called the fundamental theorem of algebra was first proved by the famous German mathematician Carl Friedrich Gauss ( ).
Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers.
The important consequences of this theorem: In general, when all real and imaginary solutions are counted (with all repeated solutions counted individually), any n th degree polynomial equation has exactly n solutions. Similarly, any n th-degree polynomial has exactly n zeros.

6. Find all the zeros of
Solutions:
The possible rational zeros are
Synthetic division or the graph can help:
Notice the real zeros appear as x-intercepts. x = 1 is repeated zero since it only “touches” the x-axis, but “crosses” at the zero x = -2.
Thus 1, 1, and –2 are real zeros. Find the remaining 2 complex zeros.

7. Find all the zeros of
Solutions:
The zeros are 1, 1, -2, and
Thus
Notice the complex zeros occurred in a conjugate pair.
The complex zeros of a polynomial functions with real coefficients always occur in complex conjugate pairs. That is, if a + bi is a zero, then a – bi must also be a zero.

8. Write a polynomial function f of least degree that has real coefficients, a leading coefficient 1, and 2 and 1 + i as zeros.
Solution:
f(x) = (x – 2)[x – (1 + i)][x – (1 – i)]

9. The rational zero theorem gives you a way to find the rational zeros of a polynomial function with integer coefficients. To find the real zeros of any polynomial functions, you may need to use a calculator.
For example, approximate the zeros of
From the screens you can see the real zeros are about –0.73 and 2.73