1. 4.6 - Fundamental Theorem of Algebra
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4.6 - Fundamental Theorem of Algebra

2. Fundamental Theorem of Algebra
Section 4.6
Pre–Calculus PreAP/Dual, Revised ©2015
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4.6 - Fundamental Theorem of Algebra

3. 4.6 - Fundamental Theorem of Algebra
Theorems
Fundamental Theorem of Algebra is the number of solutions in a polynomial equation with rational coefficients is equal to the degree of the polynomial when including all complex solutions and solutions of multiplicity. That includes all real, imaginary, and multiplicity roots.
Conjugate Root Theorem: All complex zeros (imaginaries and radical) come in conjugate pairs. If a zero is given, use the conjugate and GO BACKWARDS.  
Every IRRATIONAL or IMAGINARY number has TWO roots
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4.6 - Fundamental Theorem of Algebra

4. 4.6 - Fundamental Theorem of Algebra
Example 1
Write the simplest polynomial function with the given zeros of 4, –1, –2
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4.6 - Fundamental Theorem of Algebra

5. 4.6 - Fundamental Theorem of Algebra
Example 2
Write the simplest polynomial function with the given zeros of 2, –2, and 0.
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4.6 - Fundamental Theorem of Algebra

6. 4.6 - Fundamental Theorem of Algebra
Your Turn
Write the simplest polynomial function with the given zeros of 3, 0, –2
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4.6 - Fundamental Theorem of Algebra

7. 4.6 - Fundamental Theorem of Algebra
Example 3
Determine the LINEAR FACTORS that has 3, 1/2, and 3/2 are zeros. No fractions or decimals are accepted.
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4.6 - Fundamental Theorem of Algebra

8. 4.6 - Fundamental Theorem of Algebra
Your Turn
Determine the LINEAR FACTORS that has –2, 1/4, and –1/2 are zeros. No fractions or decimals are accepted.
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4.6 - Fundamental Theorem of Algebra

9. 4.6 - Fundamental Theorem of Algebra
Example 4
Find a second degree polynomial function with real coefficients that has 𝟐 as a zero.
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4.6 - Fundamental Theorem of Algebra

10. 4.6 - Fundamental Theorem of Algebra
Example 5
Find a fourth degree polynomial function with real coefficients that has –1, –1, and 3i are zeros.
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4.6 - Fundamental Theorem of Algebra

11. 4.6 - Fundamental Theorem of Algebra
Your Turn
Find a fifth degree polynomial function with the given zeros of +1, –2 and –2i
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4.6 - Fundamental Theorem of Algebra

12. 4.6 - Fundamental Theorem of Algebra
Example 6
Write the simplest function with zeros of 2 + i and 𝟑
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4.6 - Fundamental Theorem of Algebra

13. 4.6 - Fundamental Theorem of Algebra
Example 6
Write the simplest function with zeros of 2 + i and 𝟑
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4.6 - Fundamental Theorem of Algebra

14. 4.6 - Fundamental Theorem of Algebra
Example 7
Write the simplest function with zeros of 2 and 3 + i
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4.6 - Fundamental Theorem of Algebra

15. 4.6 - Fundamental Theorem of Algebra
Example 8
Write the simplest function with zeros of 1 – 3i, 3, and –2
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4.6 - Fundamental Theorem of Algebra

16. 4.6 - Fundamental Theorem of Algebra
Your Turn
Write the simplest function with zeros of 1 with a multiplicity of –3 and 2 – i
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4.6 - Fundamental Theorem of Algebra

17. 4.6 - Fundamental Theorem of Algebra
Finding Zeros
Use the rational root theorem (P(x) over Q(x)) to make a list of potential answers.
DIVIDE the function using
Do SYNTHETIC DIVISION or LONG DIVISION using these zeros to until you got it down to a QUADRATIC equation.
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4.6 - Fundamental Theorem of Algebra

18. 4.6 - Fundamental Theorem of Algebra
Example 9
Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero of f.
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4.6 - Fundamental Theorem of Algebra

19. 4.6 - Fundamental Theorem of Algebra
Example 9
Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero of f.
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4.6 - Fundamental Theorem of Algebra

20. 4.6 - Fundamental Theorem of Algebra
Example 9
Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero of f.
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4.6 - Fundamental Theorem of Algebra

21. 4.6 - Fundamental Theorem of Algebra
Example 10
Find all the zeros of f(x) = x4 – 5x3 + 4x2 + 2x – 8 given that 1 + i is a zero of f.
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4.6 - Fundamental Theorem of Algebra

22. 4.6 - Fundamental Theorem of Algebra
Example 11
Find all the zeros of f(x) = x3 – 4x2 + 21x – 34 given that 1 + 4i is a zero of f.
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4.6 - Fundamental Theorem of Algebra

23. 4.6 - Fundamental Theorem of Algebra
Your Turn
Find all the zeros of f(x) = x3 – 11x2 + 41x – 51 given that 4 – i is a zero of f.
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4.6 - Fundamental Theorem of Algebra

24. 4.6 - Fundamental Theorem of Algebra
Assignment
Page 313: 7, 8, all, odd, EOO
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4.6 - Fundamental Theorem of Algebra