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Algebra II: Polynomials

Problems

Complex Zeros and the Fundamental Theorem of Algebra

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Problem : What is the multiplicity of each root of P(x) = (x - 3)2(x - 5)3(x + 1) ?

x = 3 has multiplicity 2.
x = 5 has multiplicity 3.
x = - 1 has multiplicity 1.

Problem : If x = 2 - 7i is a root of P(x) , what is another root? Name one real factor of P(x) .

Root: x = 2 + 7i .
Real factor: x 2 - 4x + 53 .

Problem : If x = 4 - 3i is a root of P(x) = x 3 -12x 2 + 57x - 100 , factor P(x) completely.

P(x) = (x - 4 + 3i)(x - 4 - 3i)(x - 4)

Problem : If 2 + 5i is a root of P(x) = x 3 -5x 2 + 33x - 29 , factor P(x) completely.

P(x) = (x - 2 + 5i)(x - 2 - 5i)(x - 1) .

Problem : How many complex roots foes 4x 5 +3x 4 -2x 3 + 1 have?

5.

Problem : How many complex roots does 3x 4 +4ix 3 +3ix 2 +4x 2 - 5ix - 2x + 5i - 7 have?

4.

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