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**Problem of the Day 1) I am thinking of four numbers such that**

The sum of all the four numbers is 31. Only one of number is odd. The highest number minus the lowest number is 7. If you subtract the middle two numbers, it equals two. There are no duplicate numbers. What four numbers i am thinking of ?

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**Fundamental Theorem of Algebra **

Objective: To be able to use the Fundamental Theorem of Algebra to find polynomial equations. TS: Demonstrating understanding of concepts Warm up: T or F: A cubic function has at least one real root. T or F: A polynomial function can have no complex solutions. T or F: A polynomial function could have only one imaginary solution. T or F: A polynomial could have root 2 as its only irrational solution.

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**Linear Factorization Theorem**

The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system. Linear Factorization Theorem If f(x) is a polynomial of degree n where n > 0, f has precisely n linear factors f(x) = an(x – c1)(x – c2)∙∙∙(x – cn) where c1, c2, …, cn are complex numbers.

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**Find a cubic polynomial with zeros of 2i and 3**

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**Find the quartic polynomial with zeros -√2 and i, which passes through (1, 6)**

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**Factoring Polynomials so they are irreducable over the rationals, reals and complex zeros.**

Factor each: a) x4 – x2 – 20 x4 – 3x3 – x2 – 12x – 20 (Hint: x2 + 4 is a factor)

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You Try: 1) If -1 – 3i is a zero of x3 + 4x2 + 14x + 20, find the other zeros Factor the following: x4 + 6x2 – 27 a) Irreducible over the rationals: b) Irreducible over the reals: c) Irreducible over the complex:

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