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Warm up

Warm up. Use Synthetic Division: 1. 3x 2 – 11x + 5 x – 4 2. 5x 5 + 3x 3 +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.

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Warm up

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  1. Warm up • Use Synthetic Division: • 1. 3x2 – 11x + 5 x – 4 2. 5x5 + 3x3 +1 x + 2

  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.

  3. 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.

  4. Ex. List the possible rational roots for the following polynomials. factors of constant: factors of lead coefficient: possible rational roots:

  5. Put in order first: factors of constant: factors of lead coefficient: possible rational roots:

  6. Let’s Try One Find the POSSIBLE roots of 5x3-24x2+41x-20=0

  7. Let’s Try One 5x3-24x2+41x-20=0

  8. 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.

  9. 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

  10. Step 3 – factors • Factors of -3 = ±3, ±1 Factors of 1 = ± 1 • Step 4 – possible roots • -3, 3, 1, and -1

  11. X X³ + X² – 3x – 3 -1 1 1 -3 -3 -3 3 1 -1 (-3)³ + (-3)² – 3(-3) – 3 = -12 (3)³ + (3)² – 3(3) – 3 = 24 3 -1 0 (1)³ + (1)² – 3(1) – 3 = -4 1 0 -3 0 (-1)³ + (-1)² – 3(-1) – 3 = 0 THIS IS YOUR ROOT BECAUSE WE ARE LOOKING FOR WHAT ROOTS WILL MAKE THE EQUATION =0 1x² + 0x -3 • Step 6 – synthetic division • Step 5 – Test each root

  12. 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

  13. 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>.

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