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Sullivan Algebra and Trigonometry: Section 12.6

Sullivan Algebra and Trigonometry: Section 12.6. Objectives of this Section Decompose P/Q, Where Q Has Only Nonrepeated Factors Decompose P/Q, Where Q Has Repeated Factors Decompose P/Q, where Q Has Only Nonrepeated Irreducible Quadratic Factors

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Sullivan Algebra and Trigonometry: Section 12.6

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  1. Sullivan Algebra and Trigonometry: Section 12.6 • Objectives of this Section • Decompose P/Q, Where Q Has Only Nonrepeated Factors • Decompose P/Q, Where Q Has Repeated Factors • Decompose P/Q, where Q Has Only Nonrepeated Irreducible Quadratic Factors • Decompose P/Q, where Q Has Only Repeated Irreducible Quadratic Factors

  2. Recall the following problem from Chapter R: In this section, the process will be reversed, decomposing rational expressions into partial fractions.

  3. A rational expression P / Q is called proper if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.

  4. CASE 1: Q has only nonrepeated linear factors. Under the assumption that Q has only nonrepeated linear factors, the polynomial Q has the form where none of the numbers ai are equal. In this case, the partial fraction decomposition of P / Q is of the form where the numbers Aiare to be determined.

  5. Solve this system using either substitution or elimination.

  6. CASE 2: Q has repeated linear factors. If the polynomial Q has a repeated factor, say, (x - a) n, n> 2 an integer, then, in the partial fraction decomposition of P / Q, we allow for the terms

  7. Solve the matrix system using Cramer’s Rule

  8. CASE 3: Q contains a nonrepeated irreducible quadratic factor. If Q contains a nonrepeated irreducible quadratic factor of the form ax2 + bx + c, then, in the partial fraction decomposition of P / Q, allow for the term where the numbers A and B are to be determined.

  9. Let x = 1: Let x = -1: -12 = -4A -4 = 4B A = 3 B = -1

  10. CASE 4: Q contains repeated irreducible quadratic factors. If the polynomial Q contains a repeated irreducible quadratic factor (ax2 + bx + c)n, n> 2, n an integer, then, in the partial fraction decomposition of P / Q, allow for the terms where the numbers Ai ,Bi are to be determined.

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