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Lesson 6-5 Theorems About Roots of Polynomial Equations
Date__________
Objective: To solve equations using the Rational Root Theorem
The Rational Root Theorem: Analyzing one or more integer coefficients of the polynomial in
the equation
Consider:
x3 – 5x2 – 2x + 24 = 0 and the equivalent equation (x + 2)(x – 3)(x – 4) = 0
The roots are ____, ____, _____. The product of these roots is _______.
All the roots are factors of which term in the original polynomial________________.
In general, if the coefficients (including the constant term) in a polynomial equation are
integers, then any integer root of the equation is a factor of the _______________term.
Consider:
1 
2 
3

24x3 – 22x2 – 5x + 6 = 0 and  x    x    x    0
2 
3 
4

The roots are ____, ____, ____
The numerators, 1, 2, 3 are all factors of the constant term, _____
The denominators 2, 3, 4 are factors of the leading coefficient, _____
Rational Root Theorem
p
If
is in simplest form and is a rational root of the polynomial equation
q
anxn + an-1xn-1 + … + a1x + a0 = 0
with integer coefficients, then p must be a factor of a0 and q must be a factor of an
Finding Rational Roots
1) List the possible rational roots of x3 + x2 – 3x – 3 = 0
2) Find the possible rational roots of x3 – 4x2 – 2x + 8 = 0
M. Murray
Using the Rational Root Theorem:
1) Find the roots of x3 – 2x2 – 5x + 10 = 0
1) List possibilities
2) Test until you
find one—
Remainder = 0
3) Write quotient
4) Solve to find
roots
3) Find the roots of
x4 + x3 + x2 – 9x – 10 = 0
2) Find the roots of 3x3 + x2 – x + 1 = 0
Wrap-Up: If a polynomial equation has integer coefficients, how can you find any rational
roots the equation might have?
Assignment:
M. Murray