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Understanding the Irrational and Imaginary Root Theorems: Roots, Multiplicities, and Graphing

This note explores the Irrational and Imaginary Root Theorems, emphasizing the significance of conjugate roots in polynomial equations with integer coefficients. It includes examples to illustrate finding missing roots and writing polynomial equations based on known zeros. Additionally, the note covers the concept of root multiplicity and provides step-by-step methods for solving polynomial equations and finding relative maxima and minima using a graphing calculator. Practice problems reinforce comprehension of these critical algebraic concepts.

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Understanding the Irrational and Imaginary Root Theorems: Roots, Multiplicities, and Graphing

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  1. Notes Day 6.2 Irrational Root Theorem Imaginary Root Theorem Factors and their multiplicity Find relative max/min on the graphing calculator

  2. Notes 6.2 Irrational Root Theorem If is an S / R / Z / x-intercept, then the conjugate __________ must ALSO be an S / R / Z / x-intercept EXAMPLE A polynomial equation with integer coefficients has 4 roots. If two of them are and , what are the other two? __________________ and _________________

  3. Write a quartic Polynomial equation that has zeros at 1, -1, and

  4. Imaginary Root Theorem If a + bi is an S / R / Z / x-intercept, then the conjugate __________ must ALSO be an S / R / Z / x-intercept EXAMPLE A polynomial equation with integer coefficients has 4 roots. If two of them are 3 – i and 2i , what are the other two? ______________________ and _____________________

  5. Write a cubic Polynomial equation that has zeros at 2 and 4+i ix -4x x2 -8x2 17x x3 -4x 16 -4i -ix 4i -2x2 16x -i2 -34

  6. Multiplicity The number of times an S / R / Z / x-intercept is ___________________ in a given polynomial equation. REPEATED EXAMPLES: name the root(s) and multiplicity y = x(x2 – 4x + 4) y = x(x-2)(x-2) Once 2 Once -1 0 Once 2 Twice Twice -4

  7. Solve the equation! The square root method works since no linear term x2+ 4 = 0 x2 = – 4

  8. Solve the equation!

  9. Solve the equation!

  10. Factor to find roots but use calculator to find relative min/max. y = 2x3 + 14x2 – 3x – 21 y-int = -21

  11. Factor to find roots but use calculator to find relative min/max. y = 3x5 + 18x4 + 27x3 Work this on back of your notes! Local minimum is: -25

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