9.9
The Fundamental
Theorem of Algebra
The Fundamental
Theorem of Algebra
• Every polynomial equation with complex
coefficients and positive degree n has exactly n
complex roots.
• You may have to count the same number more than
double
once if it is a
root.
• Theorem: If a polynomial equation with real coefficients has
a  bi as a root (a and b real), then a  bi is also a
root. In other words, imaginary roots come in pairs complex conjugates
.
• Just like we did with quadratic equations, we can also write
the equation of any polynomial from its roots.
Find the polynomial equation of least
degree having the given roots.
1.
2, 1, -4
Find the polynomial equation of least
degree having the given roots.
2.
2, 1,
1 i
Given the following root(s) for the
polynomial equation, find the
remaining roots.
3.
x  3x  x  3  0; 3
3
2
Given the following root(s) for the
polynomial equation, find the
remaining roots.
4.
x 3  2x 2  4x  16  0;  1  i 3
Given the following root(s) for the
polynomial equation, find the
remaining roots.
5.
x 4  11x 2  18  0;  3i
Given the following root(s) for the
polynomial equation, find the
remaining roots.
6.
x 4  4x 3  2x 2  12x  15  0;  2  i