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PRECALCULUS I

PRECALCULUS I. Complex Numbers Real numbers + Imaginary numbers. Dr. Claude S. Moore Danville Community College. Definition of. The square root of a negative real number is not a real number. Thus, we introduce imaginary numbers by letting i =

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PRECALCULUS I

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  1. PRECALCULUS I Complex Numbers Real numbers + Imaginary numbers Dr. Claude S. MooreDanville Community College

  2. Definition of The square root of a negative real number is not a real number. Thus, we introduce imaginary numbers by letting i = So i 2 = -1, i 3 = - i , and i 4 = +1.

  3. Example: Simplifying i n Since i 2 = -1, i 3 = - i , and i 4 = +1, a simplified answer should contain no exponent of i larger than 1. Example: i 21 = i 20 i 1 = (+1)( i ) = i Example: i 35 = i 32 i 3 = (+1)( - i ) = - i NOTE: 21/4 = 5 with r = 1 and35/4 = 8 with r = 3.

  4. Definition of Complex Number For real numbers a and b,the number a + bi is a complex number. If a = 0 and b 0, the complex number bi is an imaginary number.

  5. Equality of Complex Numbers Two complex numbers a + bi and c + di, written in standard form, are equal to each other a + bi = c + di if and only if (iff) a = c and b = d.

  6. Example: Equality If (a + 7) + bi = 9- 8i, find a and b. Since a + bi = c + di if and only if (iff) a = c and b = d, a + 7 = 9 and b = -8. Thus, a = 2 and b = -8.

  7. Addition & Subtraction: Complex Numbers Two complex numbers a + bi and c + di are added (or subtracted) by adding (or subtracting) real number parts and imaginary coefficients, respectively. (a + bi ) + (c + di ) = (a + c) + (b + d )i (a + bi ) - (c + di ) = (a - c) + (b - d )i

  8. Example: Addition & Subtraction (3 + 2i ) + (-7 - 5i ) = (3 + -7) + (2 + -5)I = -4 - 3i (-6 + 9i ) - (4 - 3i ) = (-6 - 4) + (9 + 3)i = -10 + 12i

  9. Complex Conjugates Each complex number of the form a + bi has a conjugate of the form a - bi . NOTE: The product of a complex number and its conjugate is a real number. (a + bi )(a - bi ) = a2 + b2.

  10. Example: Complex Conjugates The conjugate of -5 + 6i is -5 - 6i The conjugate of 4 + 3i is 4 - 3i Recall: The product of a complex number and its conjugate is a real number. (a + bi )(a - bi ) = a2 + b2. (-5 + 6i )(-5 - 6i ) = (-5)2 + (6)2 = 25 + 36 = 41

  11. Principal Square Root of Negative If a is a positive number, the principal square root of the negative number -a is defined as Example:

  12. PRECALCULUS I Fundamental Theorem of Algebra Dr. Claude S. MooreDanville Community College

  13. The Fundamental Theorem If f (x) is a polynomial of degree n, where n > 0, then f has at least one root (zero) in the complex number system.

  14. Linear Factorization Theorem If f (x) is a polynomial of degree n where n > 0, then f has precisely n linear factors in the complex number system.

  15. Linear Factorization continued where c1, c2, … , cn are complex numbers and an is leading coefficient of f(x).

  16. Complex Roots in Conjugate Pairs Let f(x) be a polynomial function with real number coefficients. If a + bi, where b  0, is a root of the f(x), the conjugate a - bi is also a root of f(x).

  17. Factors of a Polynomial Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients where the quadratic factors have no real roots.

  18. PRECALCULUS I RATIONAL FUNCTIONS Dr. Claude S. MooreDanville Community College

  19. RATIONAL FUNCTIONS FRACTION OF TWO POLYNOMIALS

  20. DOMAIN DENOMINATOR CAN NOTEQUAL ZERO

  21. VERTICAL LINE x = a if ASYMPTOTES • HORIZONTAL LINE y = b if

  22. ASYMPTOTES OF RATIONAL FUNCTIONS If q(x) = 0, x = a is VERTICAL. HORIZONTALS: If n < m, y = 0. If n = m, y = an/bm. NO HORIZONTAL: If n > m.

  23. SLANT ASYMPTOTES OF RATIONAL FUNCTIONS If n = m + 1, then slant asymptote is y = quotient when p(x) is divided by q(x) using long division.

  24. GUIDELINES FOR GRAPHING 1. Find f(0)for y-intercept. 2. Solve p(x) = 0 to find x-intercepts. 3. Solve q(x) = 0 to find vertical asymptotes. 4. Find horizontal or slant asymptotes. 5. Plot one or more points between and beyond x-intercepts and vertical asymptote. 6. Draw smooth curves where appropriate.

  25. IMPORTANT NOTES 1.Graph will not cross vertical asymptote. f(x) = 2x / (x - 2) When q(x) = 0, f(x) is undefined. 2. Graph may cross horizontal asymptote. f(x) = 5x / (x2 + 1) 3.Graph may cross slant asymptote. f(x) = x3 / (x2 + 2)

  26. EXAMPLE 1 1.Graph will not cross vertical asymptote. f(x) = 2x / (x - 2) When q(x) = 0, f(x) is undefined. If q(x) = 0, x = a is VERTICAL asymptote.q(x) = x - 2 = 0 yields x = 2 V.A.

  27. EXAMPLE 1: Graph 1.Graph will not cross vertical asymptote. VERTICAL asymptote:q(x) = x - 2 = 0 yields x = 2 V.A. Graph off(x) = 2x / (x - 2)

  28. EXAMPLE 2 2. Graph may cross horizontal asymptote. f(x) = 5x / (x2 + 1) If n < m, y = 0 is HORIZONTAL asymptote. Since n = 1 is less than m = 2, the graph of f(x) has y = 0 as H.A.

  29. EXAMPLE 2: Graph 2.Graph may cross horizontal asymptote. If n < m, y = 0 is HORIZONTAL asymptote. Graph of f(x) = 5x / (x2 + 1)

  30. EXAMPLE 3 3.Graph may cross slant asymptote. f(x) = x3 / (x2 + 2) Recall how to find a slant asymptote.

  31. SLANT ASYMPTOTES OF RATIONAL FUNCTIONS If n = m + 1, then slant asymptote is y = quotient when p(x) is divided by q(x) using long division.

  32. EXAMPLE 3 continued 3.Graph may cross slant asymptote. f(x) = x3 / (x2 + 2) Since n = 3 is one more than m = 2, the graph of f(x) has a slant asymptote. Long division yields y = x as S.A.

  33. EXAMPLE 3: Graph 3.Graph may cross slant asymptote. Long division yields y = x as S.A. Graph of f(x) = x3 / (x2 + 2)

  34. PRECALCULUS I PARTIAL FRACTIONS Dr. Claude S. MooreDanville Community College

  35. Test 2, Wed., 10-7-98 No Use of Calculators on Test.

  36. Test 2, Wed., 10-7-98 1. Use leading coefficient test. 2. Use synthetic division. 3. Use long division. 4. Write polynomial given roots. 5. List, find all rational roots. 6. Use Descartes’s Rule of Signs. 7. Simplify complex numbers.

  37. Test 2 (continued) 8. Use given root to find all roots. 9. Find horizontal & vertical asymptotes. 10. Find x- and y-intercepts. 11. Write partial fraction decomposition. 12. ? 13. ? 14. ?

  38. PARTIAL FRACTIONS RATIONAL EXPRESSION EQUALS SUM OF SIMPLER RATIONAL EXPRESSIONS

  39. DECOMPOSTION PROCESS IF FRACTION IS IMPROPER, DIVIDE AND USE REMAINDER OVER DIVISOR TO FORM PROPER FRACTION.

  40. FACTOR DENOMINATOR COMPLETELY FACTOR DENOMINATOR INTO FACTORS AS LINEAR FORM: (px + q)m and QUADRATIC: (ax2 + bx + c)n

  41. EXAMPLE 1 Change improper fraction to proper fraction. Use long division and write remainder over the divisor.

  42. EXAMPLE 1 continued Find the decomposition of the proper fraction.

  43. EXAMPLE 1 continued • Completely factor the denominator. • Write the proper fraction as sum of fractions with factors as denominators.

  44. EXAMPLE 1 continued Multiply by LCD to form basic equation: x - 1 = A(x + 3) + B(x + 1)

  45. GUIDELINES FOR LINEAR FACTORS 1. Substitute zeros of each linear factor into basic equation. 2. Solve for coefficients A, B, etc. 3. For repeated factors, use coefficients from above and substitute other values for x and solve.

  46. EXAMPLE 1 continuedSolving Basic Equation To solve the basic equation: Let x = -3 and solve for B = 2. Let x = -1 and solve for A = -1.

  47. EXAMPLE 1 continued Since A = - 1 and B = 2, the proper fraction solution is

  48. EXAMPLE 1 continued Thus, the partial decomposition of the improper fraction is as shown below.

  49. EXAMPLE 1: GRAPHS The two graphs are equivalent.

  50. EXAMPLE 2 Find the partial fraction decomposition of the rational expression:

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