5.5 Theorems about Roots of
Polynomial Equations
P312-315
So this means that the POSSIBLE
RATIONAL ROOTS have the form
Factor of constant term
Factor of leading coefficient
Ex) What are the rational roots of
15 x  32 x  3 x  2  0 ?
3
2
Steps….
Find the possible rational roots.
Test the possible rational roots to find a root.
Factor polynomial by using synthetic division.
Use factored polynomial to find other roots.
Ex) What are the rational roots of
15 x  32 x  3 x  2  0 ?
3
2
This means that
Irrational roots come in conjugate pairs
Imaginary roots come in conjugate pairs
Ex) A quartic polynomial P(x) has rational
coefficients. If 2 and 1  i
are roots of P(x)=0, what are the other
roots?
Ex) What is a third degree polynomial
function y = P(x) with rational coefficients
so that P(x)=0 has the roots  4 and
2i ?
Ex) What does Descarte’s Rule of Signs
tell you about the real roots of
3
2
8 x  2 x  14 x  5
?