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Page ) Origin 21) 23) Y-axis, Origin 25) Odd 27) Even 29) 31)

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Presentation on theme: "Page ) Origin 21) 23) Y-axis, Origin 25) Odd 27) Even 29) 31)"— Presentation transcript:

1 Page 189 19) Origin 21) 23) Y-axis, Origin 25) Odd 27) Even 29) 31)
33) Neither 2/28/2019 7:54 PM 4.1A - Related Theorems

2 Pre–Calculus PreAP/Dual, Revised ©2015
Rational Zero Test Section 4.2 Pre–Calculus PreAP/Dual, Revised ©2015 2/28/2019 7:54 PM 4.1A - Related Theorems

3 Steps Identify the constant term and coefficient of the polynomial
List all multiples of each number and don’t forget the plus-or-minus symbol for each polynomial Apply the equation: 𝒑 𝒒 = Factors of Constant Term Factors of Leading Coefficient List ALL of the possible rational zeros 2/28/2019 7:54 PM 4.1A - Related Theorems

4 What multiplies with what equals that number?
Example 1 List all possible rational zeros for x8 + 18x4 – 9x2 + 2 = 0 What multiplies with what equals that number? 2/28/2019 7:54 PM 4.1A - Related Theorems

5 Example 2 List all possible rational zeros for f(x) = 2x4 + x3 – 17x2 – 4x + 6 2/28/2019 7:54 PM 4.1A - Related Theorems

6 Your Turn Find ALL POSSIBLE rational zeros for 4x4 – 21x3 + 18x2 + 19x – 6 = 0 2/28/2019 7:54 PM 4.1A - Related Theorems

7 Finding Rational Zeros
Where the graphs cross the x–axis (y is zero) Known as zeros or roots To find the real–number zeros, it can be rational, irrational and/or imaginary numbers To find out how many zeros the graph has, the easiest way to find the amount of degrees the graphs has 2/28/2019 7:54 PM 4.1A - Related Theorems

8 Steps Determine what the leading coefficient and constant term are.
Put it in the rational zero test (p/q) Use synthetic division to find which factor gives the remainder of zero. If the polynomial cannot be factored, repeat steps 2 and 3 again Factor or use the Quadratic Formula to get the leftover polynomial out. Equal it all to zero and put it in x= form. If possible, check in graphing calculator. 2/28/2019 7:54 PM 4.1A - Related Theorems

9 Example 3 Find the real and imaginary rational zeros for f(x) = x3 + x2 – 2x – 2 Step 1: What are the factors of the constant term and the leading coefficient? Constant Term (the number without the letter): Coefficient: (the number in front of the letter with the highest degree) Step 2: Put it in rational–zero test 2/28/2019 7:54 PM 4.1A - Related Theorems

10 Example 3 Find the real and imaginary rational zeros for f(x) = x3 + x2 – 2x – 2 Step 3: Test each possible rational form 2/28/2019 7:54 PM 4.1A - Related Theorems

11 Example 3 Find the real and imaginary rational zeros for f(x) = x3 + x2 – 2x – 2 Step 4: Factor or use the Quadratic Formula to get the leftover polynomial out. 2/28/2019 7:54 PM 4.1A - Related Theorems

12 Example 3 Find the real and imaginary rational zeros for f(x) = x3 + x2 – 2x – 2 Step 5: Equal it all to zero and put it in x = form. 2/28/2019 7:54 PM 4.1A - Related Theorems

13 Example 3 Find the real and imaginary rational zeros for f(x) = x3 + x2 – 2x – 2 Step 6: Use graphing calculator to check 2/28/2019 7:54 PM 4.1A - Related Theorems

14 Example 4 Find the real and imaginary rational zeros for f(x) = 3x3 + 7x2 – 22x – 8 2/28/2019 7:54 PM 4.1A - Related Theorems

15 Example 5 Find the real and imaginary rational zeros for f(x) = x3 – 7x2 + 11x – 5 2/28/2019 7:54 PM 4.1A - Related Theorems

16 Your Turn Find the real and imaginary rational zeros for f(x) = 6x3 + x2 – 10x + 3 2/28/2019 7:54 PM 4.2 – Real Zeros

17 Example 6 Find the real and imaginary rational zeros for f(x) = 2x4 + x3 – 17x2 – 4x + 6 Step 1: What are the factors of the constant term and the leading coefficient? Step 2: Apply p/q 2/28/2019 7:54 PM 4.1A - Related Theorems

18 Example 6 Find the real and imaginary rational zeros for f(x) = 2x4 + x3 – 17x2 – 4x + 6 Step 3: Test each possible rational form 2/28/2019 7:54 PM 4.2 – Real Zeros

19 Example 6 Find the real and imaginary rational zeros for f(x) = 2x4 + x3 – 17x2 – 4x + 6 Step 3: Test each possible rational form 2/28/2019 7:54 PM 4.2 – Real Zeros

20 Example 6 Find the real and imaginary rational zeros for f(x) = 2x4 + x3 – 17x2 – 4x + 6 Step 3: Test each possible rational form 2/28/2019 7:54 PM 4.2 – Real Zeros

21 Example 6 Find the real and imaginary rational zeros for f(x) = 2x4 + x3 – 17x2 – 4x + 6 Step 4: Identify the equation & Solve 2/28/2019 7:54 PM 4.2 – Real Zeros

22 Example 6 Find the real and imaginary rational zeros for f(x) = 2x4 + x3 – 17x2 – 4x + 6 Step 4: Identify the equation & Solve 2/28/2019 7:54 PM 4.2 – Real Zeros

23 Example 6 Find the real and imaginary rational zeros for f(x) = 2x4 + x3 – 17x2 – 4x + 6 Step 5: Check 2/28/2019 7:54 PM 4.2 – Real Zeros

24 Example 7 Find the real and imaginary rational zeros for f(x) = x4 –x3 - x2 – x – 2 2/28/2019 7:54 PM 4.1A - Related Theorems

25 Assignment Worksheet 2/28/2019 7:54 PM 4.1A - Related Theorems

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