6-5 and 6-6 Theorems About Roots of Polynomial Equations
How do we solve x 3  5 x 2  2 x  24  0 ? If we knew the factored form, ( x  2)( x  3)( x  4)  0 , we could find the
zeros of the polynomial. What if we don’t know how to factor the polynomial?
Compare these zeros to the factors of 24, the constant term of our polynomial.
Rational Root Theorem:
If the leading coefficient is 1, then the roots will all be integer factors of the constant term. They can be either positive or
negative.
If the leading coefficient isn’t 1, our roots will be rational numbers. The numerators will be factors of the constant term
and the denominators will be factors of our leading coefficient.
Find possible rational roots, then graph to see which possible root is actually a root:
x 3  x 2  3x  3  0
3x 3  x 2  15 x  5  0
2x3  x 2  2x  1  0
After finding the root we can divide it out. If our remaining expression is a quadratic expression, we can use factoring or
the Quadratic Formula to find the other two roots.
Definition:
Conjugates:
Complex Conjugates:
Irrational Root Theorem:
Imaginary Root Theorem:
Using the given roots, find additional roots:
Roots: 1  3 and  11
Roots: 2  7 and
Roots: 3  i and 2i
Roots: 3i and  2  i
5
Roots: 2  5 and
7
Roots: Roots: 2  9i and 7i
Fundamental Theorem of Algebra (Corollary):
x 4  3x 3  x 2  x  3  0
Number of complex roots:
Maximum number of real roots:
Remember, imaginary roots come in pairs!
x3  2x 2  4x  6  0