Chapter 8 Notes

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Theorems about Roots• Number of Roots Theorem: Every polynomial withcomplex coefficients and positive degree n has exactly nroots.• Conjugate Root Theorem: If a polynomial equation withreal coefficients has a + bi as a root (a & b ∈R, b ≠ 0),then a - bi is also a root.• Descartes’ Rule of Signs: Let P(x) be a simplifiedpolynomial with real coefficients and terms arranged indecreasing degree of x.– The number of positive real roots of P(x) = 0 equals thenumber of variations of sign of P(x) or is fewer thanthis number by an even integer.– The number of negative real roots of P(x) = 0 equals thenumber of variations of sign of P(- x) or is fewer thanthis number by an even integer.
Steps for Homework Problems• Problems 1-4: write each number as a factor and theconjugate of any complex number as a factor, thenmultiply the factors together.• Problems 5-8: Write the conjugate of the complex root.• Problems 9-12: synthetically divide by the given complexnumber and its conjugate. There is an easier way to do thisproblem if you under stand polynomial long division.• Problems 13-20: use Descartes’ Rule of Signs todetermine the possible number of each type of roots.• Problems 21 & 22: Write the given numbers and theirconjugates as factors. Multiply these factors together.• Problems; 23 & 24: Perform synthetic division with thegiven complex number and its conjugate.• Problem 26: use Descartes’ Rule of Signs to help yougenerate a table for all possible roots.
Page 1 and 2: Section 8-1Direct Variation and Pro
Page 3 and 4: Direct Variation• A linear functi
Page 5 and 6: Section 8-2Inverse and Joint Variat
Page 7 and 8: Inverse Variation• a function def
Page 9 and 10: Example Homework Problem• Suppose
Page 11 and 12: Objectives• to perform polynomial
Page 13 and 14: Steps to perform polynomial long di
Page 15 and 16: Example of polynomial long division
Page 17 and 18: Section 8-4Synthetic Division
Page 19 and 20: Synthetic Division• Synthetic div
Page 21 and 22: Example of synthetic division2x32
Page 23 and 24: Objectives• to use synthetic subs
Page 25 and 26: Steps for homework problems• Prob
Page 27 and 28: Example for Problems 1-8P(x) = x 3
Page 29 and 30: Example for Problems 17-20x 3 + 3x
Page 31 and 32: Example for Problems 29-32x 4 -3x 3
Page 33 and 34: Example for Problems 35-37P(x) = x
Page 35: Objectives• to write a polynomial
Page 39 and 40: Example for Problems 5-8x 3 -3x 2 +
Page 41 and 42: Example for Problems 13-20• x 4 +
Page 43 and 44: Example for Problems 23 & 24x 3 + 2
Page 45 and 46: Objectives• to find the rational
Page 47 and 48: Steps for the homework• Use Desca
Page 49 and 50: • 2x 2 - 4x + 4 = 0: has a GCF of
Page 51 and 52: Objectives• to approximate to the
Page 53 and 54: Steps for homework• Perform Desca
Page 55 and 56: 1 1 -3 0 0 51 -2 -2 -21 -2 -2 -2 32
Page 57 and 58: Objectives• to approximate values
Page 59: Example Homework Problem• Approxi

Chapter 8 Notes

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