Download Rational Root Theorem

Roots & Zeros of
Polynomials III
Using the Rational Root Theorem
to Predict the Rational Roots of a
Polynomial
Created by K. Chiodo, HCPS
Find the Roots of a Polynomial
For higher degree polynomials, finding the complex
roots (real and imaginary) is easier if we know one
of the roots.
Descartes’ Rule of Signs can help get you started.
Complete the table below:
Polynomial
y  x  2x  3
4
2
y  x  7x 17x 15
3
2
y  3x  x  3x  x 1
4
3
2
# + Real # - Real # Imag.
Roots
Roots
Roots
The Rational Root Theorem
The Rational Root Theorem gives us a tool to predict
the Values of Rational Roots:
If P(x)  a0 x n  a1 x n1  ...  an 1 x  an , where the
coeffiecients are all integers,
p
& a rational zero of P(x) in reduced form is
, then
q
 p must be a factor of an (the constant term) &
 q must be a factor of a0 (the leading coefficient).
List the Possible Rational Roots
For the polynomial:
All possible values of:
f (x)  x  3x  5x  15
3
2
p: 1, 3,  5
q: 1
All possible Rational Roots of the form p/q:
p
:  1,  3,  5
q
Narrow the List of Possible Roots
For the polynomial:
f (x)  x  3x  5x  15
Descartes’ Rule:
# + Real Roots = 3 or 1
3
2
#  Real Roots = 0
# Imag. Roots = 2 or 0
All possible Rational Roots of the form p/q:
p
: 1, 3, 5
q
Find a Root That Works
For the polynomial:
f (x)  x  3x  5x  15
3
2
Substitute each of our possible rational roots into f(x).
If a value, a, is a root, then f(a) = 0. (Roots are
solutions to an equation set equal to zero!)
f (1)  1 3  5  15  12
f (3)  27  27  15  15  0
f (5)  125  75  25  15  60
*
Find the Other Roots
Now that we know one root is x = 3, do the other two
roots have to be imaginary? What other category have
we left out?
To find the other roots, divide the factor that we know
into the original polynomial:

x  3 x  3x  5x  15
3
2
Find the Other Roots (con’t)
x2  5
3
2
x  3 x  3x  5x  15

The resulting polynomial is a quadratic, but it doesn’t
have real factors. Solve the quadratic set equal to zero
by either using the quadratic formula, or by isolating the
x and taking the square root of both sides.
Find the Other Roots (con’t)
The solutions to the
quadratic equation:
x  i 5, i 5
For the polynomial:
f (x)  x  3x  5x  15
The three complex
roots of the
polynomial are:
x  3, i 5,  i 5
3
2
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