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Warm up
• Use Synthetic Division:
• 1. 3x2 – 11x + 5
x–4
2. 5x5 + 3x3 +1
x+2
Lesson 4-4 Rational Root
Theorem
Objective: To use the rational root
theorem to determine the number of
possible rational roots in a polynomial.
Rational Roots Theorem:
If a polynomial equation has a rational
root, then this root is one of the possible
quotients of a factor of the constant term,
divided by a factor of the leading
coefficient.
factor of constant
rational root 
factor of lead coefficient
Integral Root Theorem
• If the leading coefficient of the polynomial is 1 then the
rational roots of the polynomial must be integers.
Ex. List the possible rational roots for the
following polynomials.
2x  3x  2x  5  0
3
2
factors of constant (p): 1, 5
factors of lead coefficient(q): 1, 2
possible rational roots
1 5
1, 5,  , 
p/q:
2 2
6x  2x  4x
9
15
30
Put in order first:
4x
15
 2x  6x  3  0
9
factors of constant (p): 1, 3
factors of lead coefficient (q): 1, 2, 4
possible rational roots(p/q):
1 3 1
3
1, 3,  ,  ,  , 
2 2
4
4
Let’s Try One
Find the POSSIBLE roots of
5x3-24x2+41x-20=0
Let’s Try One
5x3-24x2+41x-20=0
That’s a lot of answers!
• Obviously 5x3-24x2+41x-20=0 does not
have all of those roots as answers.
• Remember: these are only POSSIBLE
roots. We take these roots and figure out
what answers actually WORK.
• Step 1 – find p and
q
• p= 3
• q= 1
• Step 2 – by RRT, the
only rational root is
of the form…
• Factors of p
Factors of q
• Step 3 – factors
• Factors of -3 = ±3, ±1
Factors of 1 = ± 1
• Step 4 – possible
roots
• -3, 3, 1, and -1
• Step 5 – Test each
root
X
-3
3
• Step 6 – synthetic
division
X³ + X² – 3x – 3
(-3)³ + (-3)² – 3(-3) – 3 = -12
-1
1
1
-3
-3
-1
0
3
0
-3
0
(3)³ + (3)² – 3(3) – 3 = 24
1
(1)³ + (1)² – 3(1) – 3 = -4
-1
(-1)³ + (-1)² – 3(-1) – 3 = 0
1
THIS IS YOUR ROOT
BECAUSE WE ARE LOOKING
FOR WHAT ROOTS WILL
MAKE THE EQUATION =0
1x² + 0x
-3
• Step 7 – Rewrite
• x³ + x² - 3x - 3
= (x + 1)(x² – 3)
• Step 8– factor more
and solve
• (x + 1)(x² – 3)
• (x + 1)(x – √3)(x + √3)
• Roots are -1, ± √3
Descartes' Rule of Signs
If f(x)  anxn  an1xn1  …  a2x2  a1x  a0 be a polynomial with real
coefficients.
1. The number of positive real zeros of f is either equal to the number
of sign changes of f(x) or is less than that number by an even integer.
If there is only one variation in sign, there is exactly one positive real
zero.
2. The number of negative real zeros of f is either equal to the number
of sign changes of f(x) or is less than that number by an even
integer. If f(x) has only one variation in sign, then f has exactly one
negative real zero.
EXAMPLE:
Using Descartes’ Rule of Signs
Determine the possible number of positive and negative real zeros of
f(x)  x3  2x2  5x + 4.
Solution
1. To find possibilities for positive real zeros, count the number of sign
changes in the equation for f(x). Because all the terms are positive, there
are no variations in sign. Thus, there are no positive real zeros.
2. To find possibilities for negative real zeros, count the number of sign
changes in the equation for f(x). We obtain this equation by replacing x
with x in the given function.
f(x)  x3  2x2  5x + 4
Replace x with x.
f(x)  (x)3  2(x)2  5(x)  4
 x3  2x2  5x + 4
This is the given polynomial function.
EXAMPLE:
Using Descartes’ Rule of Signs
Determine the possible number of positive and negative real zeros of
f(x)  x3  2x2  5x + 4.
Solution
Now count the sign changes.
f(x)  x3  2x2  5x + 4
1
2
3
There are three variations in sign.
# of negative real zeros of f is either equal to 3, or is less than this number by
an even integer.
This means that there are either 3 negative real zeros
or 3  2  1 negative real zero.
Practice
• Find the number of possible positive real roots and the
possible negative real zeros for
f ( x)  2 x  x  2 x  5 x  1
4
3
2
Sources
• Ponderosa High School Math Department. Ponderosa
High School, n.d. Web. 21 Jan. 2013.
<http://phsmath.org>.
• "6.5 Theorems About Roots of Polynomial Equations."
Pleasanton Unified School District. N.p., n.d. Web. 21
Jan. 2013. <http://www.pleasanton.k12.ca.us>.
• University of West Georgia. University of West
Georgia, Aug. 2005. Web. 2 Jan. 2014.
<http://www.westga.edu/~srivera/Mat%201111hp.htm>.