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>.