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

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Transcript of 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

Warm upUse Synthetic Division:1. 3x2 11x + 5 x 4

2. 5x5 + 3x3 +1 x + 23x + 1 + (9/(x-4))5x^4 10x^3 +23x^2 - 46x + 92 -(183/(x+2))

1Lesson 4-6 Rational Root TheoremObjective: 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.

Ex. List the possible rational roots for thefollowing 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:Lets Try OneFind the POSSIBLE roots of 5x3-24x2+41x-20=0Lets Try One5x3-24x2+41x-20=0

Thats 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 = -3q = 1Step 2 by RRT, the only rational root is of the formFactors of pFactors of q

Step 3 factors

Factors of -3 = 3, 1Factors of 1 = 1Step 4 possible roots

-3, 3, 1, and -1

Step 5 Test each root

Step 6 synthetic division

X X + X 3x 3 -3

3

1

-1(-3) + (-3) 3(-3) 3 = -12 (3) + (3) 3(3) 3 = 24 (1) + (1) 3(1) 3 = -4 (-1) + (-1) 3(-1) 3 = 0 THIS IS YOUR ROOT BECAUSE WE ARE LOOKINGFOR WHAT ROOTS WILL MAKE THE EQUATION =0-11 1 -3 -301-330-101x + 0x -3

Step 7 Rewrite

x + x - 3x - 3 = (x + 1)(x 3) Step 8 factor more and solve

(x + 1)(x 3)(x + 1)(x 3)(x + 3)

Roots are -1, 3

SourcesPonderosa High School Math Department. Ponderosa High School, n.d. Web. 21 Jan. 2013. .

"6.5 Theorems About Roots of Polynomial Equations." Pleasanton Unified School District. N.p., n.d. Web. 21 Jan. 2013. .