Math 3.3 Notes Rational Root Theorem

Advanced Math Section 3.3 Notes Rational Root Theorem Name: Learning Target:  Finding the zeros of a polynomial using the Rational Root Theorem. Warm Up: Use synthetic division to see if 𝑐 is a zero of 𝑃(𝑥). a. 𝑃(𝑥) = 4𝑥 3 − 10𝑥 2 − 8𝑥 + 6, 𝑐=1 b. 𝑃(𝑥) = 𝑥 4 − 2𝑥 2 − 100𝑥 − 75, 𝑐=5 A polynomial function 𝑃 of degree 𝑛 has a most _______ zeros, including ________________________________. Rational Zero Theorem: We will do an actual example to show this theorem! Given 𝑃(𝑥) = 𝑥 3 + 2𝑥 2 − 𝑥 − 2 𝑝 = _________ (________________________) 𝑞 = _________ (________________________) Factors of 𝑝: Factors of 𝑞:  Possible rational zeros is the following list:  All possible_________ _______________________ divided by all possible ________________________________  Complete the list by putting a ________ sign in front of each possible rational zero List of possible rational zeros from the example above: Steps to Finding Real Zeros of 𝑷(𝒙). 1. Do _______________________________________ using the list you just generated to find the rational zeros. 2. Once you get the quotient down to a ____________________, ______________________ to solve for the remaining rational zeros. 3. If the quadratic cannot be factored, the remaining zeros are either ____________________________ or _________________________ and should be found using the ____________________ ____________________. 4. Once you find a rational zero, if what’s left is still greater than a quadratic, you should always check for _____________________________________. 5. The ___________________ equals the ________________________ of zeros!! Turn the page over to continue the example for the Rational Root Theorem Example 1: Given 𝑃(𝑥) = 𝑥 3 + 2𝑥 2 − 𝑥 − 2 [This was copied for you from the previous page!] Degree = __________ Number of Zeros = _________ 𝑝=𝟐 𝑞=𝟏 Factors of 𝑝: 𝟏, 𝟐 Factors of 𝑞: 𝟏 Possible rational zeros is the list of all possible factors of 𝑝 divided by all possible factors of 𝑞. Put a ± sign in front of each possible rational zero! List of possible rational zeros: ± 𝟏, ± 𝟐 Start the synthetic division with +𝟏 Example 2: Given 𝑃(𝑥) = 𝑥 3 + 2𝑥 2 − 5𝑥 − 6 What are the rational zeros of 𝑃(𝑥)?___________________ Degree = __________ Number of Zeros = _________ 𝑝 = __________ 𝑞 = ________ Factors of 𝑝: Factors of 𝑞: Possible rational zeros is the list of all possible factors of 𝑝 divided by all possible factors of 𝑞. Put a ± sign in front of each possible rational zero! List of possible rational zeros: Start the synthetic division with +1 What are the rational zeros of 𝑃(𝑥)?___________________ Example 3: Given 𝑃(𝑥) = 𝑥 4 + 4𝑥 3 − 𝑥 2 − 16𝑥 − 12 Degree = __________ Number of Zeros = _________ 𝑝 = __________ 𝑞 = ________ Factors of 𝑝: Factors of 𝑞: Possible rational zeros is the list of all possible factors of 𝑝 divided by all possible factors of 𝑞. Put a ± sign in front of each possible rational zero! List of possible rational zeros: Start the synthetic division with +1 Example 4: Given 𝑃(𝑥) = 2𝑥 3 + 9𝑥 2 + 10𝑥 + 3 What are the rational zeros of 𝑃(𝑥)?___________________ Degree = __________ Number of Zeros = _________ 𝑝 = __________ 𝑞 = ________ Factors of 𝑝: Factors of 𝑞: Possible rational zeros is the list of all possible factors of 𝑝 divided by all possible factors of 𝑞. Put a ± sign in front of each possible rational zero! List of possible rational zeros: Start the synthetic division with +1 Key Ideas: What are the rational zeros of 𝑃(𝑥)?___________________ Advanced Math 1. Section 3.3 HW Rational Root Theorem Given 𝑃(𝑥) = 𝑥 3 + 3𝑥 2 − 6𝑥 − 8 Name: Degree = __________ Number of Zeros = _________ 𝑝 = __________ 𝑞 = ________ Factors of 𝑝: Factors of 𝑞: Possible rational zeros is the list of all possible factors of 𝑝 divided by all possible factors of 𝑞. Put a ± sign in front of each possible rational zero! List of possible rational zeros: Start the synthetic division with +1 2. Given 𝑃(𝑥) = 𝑥 3 − 3𝑥 − 2 What are the rational zeros of 𝑃(𝑥)?___________________ Degree = __________ Number of Zeros = _________ 𝑝 = __________ 𝑞 = ________ Factors of 𝑝: Factors of 𝑞: Possible rational zeros is the list of all possible factors of 𝑝 divided by all possible factors of 𝑞. Put a ± sign in front of each possible rational zero! List of possible rational zeros: Start the synthetic division with +1 Don’t forget to put a zero where 𝒙𝟐 belongs! What are the rational zeros of 𝑃(𝑥)?___________________ 3. Given 𝑃(𝑥) = 2𝑥 3 + 9𝑥 2 − 2𝑥 − 9 Degree = __________ Number of Zeros = _________ 𝑝 = __________ 𝑞 = ________ Factors of 𝑝: Factors of 𝑞: Possible rational zeros is the list of all possible factors of 𝑝 divided by all possible factors of 𝑞. Put a ± sign in front of each possible rational zero! List of possible rational zeros: Start the synthetic division with +1 4. Given 𝑃(𝑥) = 2𝑥 4 + 3𝑥 3 − 4𝑥 2 − 3𝑥 + 2 What are the rational zeros of 𝑃(𝑥)?___________________ Degree = __________ Number of Zeros = _________ 𝑝 = __________ 𝑞 = ________ Factors of 𝑝: Factors of 𝑞: Possible rational zeros is the list of all possible factors of 𝑝 divided by all possible factors of 𝑞. Put a ± sign in front of each possible rational zero! List of possible rational zeros: Start the synthetic division with +1 What are the rational zeros of 𝑃(𝑥)?___________________ 5. Given 𝑃(𝑥) = 𝑥 3 − 19𝑥 − 30 Degree = __________ Number of Zeros = _________ 𝑝 = __________ 𝑞 = ________ Factors of 𝑝: Factors of 𝑞: Possible rational zeros is the list of all possible factors of 𝑝 divided by all possible factors of 𝑞. Put a ± sign in front of each possible rational zero! List of possible rational zeros: Start the synthetic division with +1 Don’t forget to put a zero where 𝒙𝟐 belongs! 6. What are the rational zeros of 𝑃(𝑥)?___________________ Given 𝑃(𝑥) = 𝑥 4 + 2𝑥 3 − 3𝑥 2 − 8𝑥 − 4 Degree = __________ Number of Zeros = _________ 𝑝 = __________ 𝑞 = ________ Factors of 𝑝: Factors of 𝑞: Possible rational zeros is the list of all possible factors of 𝑝 divided by all possible factors of 𝑞. Put a ± sign in front of each possible rational zero! List of possible rational zeros: Start the synthetic division with +1 Don’t forget to check for multiplicities! Ex 1: 1, − 1, − 2 Ex 2: What are the rational zeros of 𝑃(𝑥)?___________________ − 1, − 3, 2 Ex 3: − 1, 2, − 2, − 3 Ex 4: − 1, − 3, − 1 2 ****************************************************************************************************************************************************************************** 1. − 1, 2, − 4 4. 1, − 1, − 2, 2. – 1 multiplicity 2, 1 2 5. −2, − 3, 5 2 3. 1, − 1, − 9 2 6. − 1 multiplicity 2, 2, −2