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

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Presentation transcript:

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

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.

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

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

Let’s Try One Find the POSSIBLE roots of 5x 3 -24x 2 +41x-20=0

Let’s Try One 5x 3 -24x 2 +41x-20=0

That’s a lot of answers! Obviously 5x 3 -24x 2 +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)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 LOOKING FOR WHAT ROOTS WILL MAKE THE EQUATION = x² + 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 "6.5 Theorems About Roots of Polynomial Equations." Pleasanton Unified School District. N.p., n.d. Web. 21 Jan