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Warm up
• Use Synthetic Division:• 1. 3x2 – 11x + 5
x – 4
2. 5x5 + 3x3 +1
x + 2
Lesson 4-6 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 rationalroot, then this root is one of the possiblequotients of a factor of the constant term,divided by a factor of the leading coefficient.
factor of constantrational root
factor of lead coefficient
Ex. List the possible rational roots for thefollowing polynomials.
3 22 3 2 5 0x x x
factors of constant: 1, 5
factors of lead coefficient: 1, 2
possible rational roots: 1 51, 5, ,
2 2
9 156 2 4 3 0x x x
15 94 2 6 3 0x x x
factors of constant: 1, 3
factors of lead coefficient: 1, 2, 4
possible rational roots:1 3 1 3
1, 3, , , ,2 2 4 4
Put in order first:
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
• Step 6 – synthetic division
X X³ + X² – 3x – 3
-3
3
1
-1
(-3)³ + (-3)² – 3(-3) – 3 = -12
(3)³ + (3)² – 3(3) – 3 = 24
(1)³ + (1)² – 3(1) – 3 = -4
(-1)³ + (-1)² – 3(-1) – 3 = 0
THIS IS YOUR ROOT BECAUSE WE ARE LOOKINGFOR WHAT ROOTS WILL MAKE THE EQUATION =0
-1 1 1 -3 -3
0
1 -3
3
0
-1
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
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>.