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Imaginary Zeros

Imaginary Zeros. Section 2.5. COMPLEX NUMBERS a + b i. Pure Imaginary “b i ”. Pure Real “a”. Rational Irrational Integers etc. Complex Number 4 + 2i. 4 real. 2i imaginary. Imaginary Numbers. i = . Re-write each as an imaginary number. 1. 2. 3.

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Imaginary Zeros

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  1. Imaginary Zeros Section 2.5

  2. COMPLEX NUMBERS a + bi Pure Imaginary “bi ” Pure Real “a” Rational Irrational Integers etc.

  3. Complex Number 4 + 2i 4 real 2i imaginary

  4. Imaginary Numbers • i=

  5. Re-write each as an imaginary number 1. 2. 3.

  6. Operating with imaginary numbers • 2i + 3i = • (2i )(3i ) = • (5 – 3i )(5 + 3i ) =

  7. Fundamental Theorem of Algebra • A polynomial of degree “n”has a TOTAL of “n” zeros (real & imaginary) Ex) f(x) = x3 – 5x + 5 Total Zeros: Real Zeros: Imaginary Zeros:

  8. Conjugates • (a + bi ) and (a – bi ) are conjugates If (a + bi ) is a root, then (a – bi ) is also

  9. Example 1: Write a polynomial function whose zeros include -2 and 5i *Must also include CONJUGATE of 5i x = -2x = 5i x = -5i

  10. Example 2: Find all zeros f(x) = x2 – 2x + 5 • Graph does not cross x-axis • Use quadratic formula to find imaginary roots

  11. Example 3: Write the polynomial as a product of linear factors f(x) = 2x3 – x2 + 2x – 3

  12. 1. Write a polynomial with zeros 4, -3, and -2i 2. Find all real & imaginary zeros f(x) = x4 + x3 + 5x2 – x – 6

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