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Warm-up 1/28/08

Warm-up 1/28/08. Find the surface area and volume of a box with one edge of unknown length, a second edge one unit longer, and a third edge one unit shorter. Write your answers as a polynomial. Surface Area = 2B + Ph S.A. = 2x(x-1)(x+1) + 2x(x-1) + 2x(x+1) S.A. = 6x 2 – 2 Volume = lwh

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Warm-up 1/28/08

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  1. Warm-up 1/28/08 Find the surface area and volume of a box with one edge of unknown length, a second edge one unit longer, and a third edge one unit shorter. Write your answers as a polynomial. Surface Area = 2B + Ph S.A. = 2x(x-1)(x+1) + 2x(x-1) + 2x(x+1) S.A. = 6x2 – 2 Volume = lwh V = x(x-1)(x+1) V = x3 - x

  2. Warm-up 1/29/08 • Graph f(x) = x4 – 2x2 – 1 • At what value(s) of x does the function f attain its minimum value? • What is the minimum value?

  3. §9.1: Polynomial Models How do you create and interpret Polynomial models? Polynomial: Expression in the form: anxn + an-1xn-1 + an-2xn-2 + a1x+ a0 Ex) a4x4 + a3x3 + a2x2 + a1x+ a0

  4. Polynomial Function – Any sum or difference of power functions and constants. Degree – The exponent determines the degree of the function. Standard form – All like terms are combined. The function is written in descending order by degree. Leading Coefficient

  5. Functions like y = x4 and w = .084C3 are power functions. Power function – A function with the form y = axn, where a≠0 and n is a positive integer. Even functions – Have the y-axis as the axis of symmetry Odd functions – have the origin as the point of symmetry. *A graph has point symmetry if there is a ½ turn (rotation of 180 deg) that maps the graph onto itself.

  6. Try these Graph each function. Describe its symmetry. Tell whether it is even, odd, or neither. • Y = x5 • Y = IxI • Y = 2x + 3 • Y = 2x4

  7. The Degree of a Polynomial • Monomial • Binomial • Trinomial • Polynomial

  8. Describing Polynomials

  9. Name & degree: 5x2 – 7x -10x3 X2 + 3x - 2 binomial, 2 Monomial, 3 Trinomial, 2 Classify each polynomial

  10. Assignment • Section 9.1 • p.560 – 562 • #1 – 5, 11 – 14, • #16 – 20 (calculator) • #22 – 24 (calculator)

  11. Warm-up 1/30/08*Use stat, edit, then cubic reg:

  12. 9.2: Graphs of Polynomials LEQ: How do you find important points on a polynomial function? Vocabulary…

  13. Important Points: • Maximum Value • Minimum Value • Extreme Values (extrema) • Relative Extremum (turning points) • Relative Maximum • Relative Minimum • Zeros (roots) – using “table”

  14. End Behavior • Positive Intervals • Negative Intervals • Increasing Interval • Slope between two points is increasing • Decreasing Interval • Slope between two points is decreasing

  15. Assignment Section 9.2 P.567-568 #1-15

  16. Warm-up 1/31/08 Solve the following system: d = 43 c – d =12 2b + 3c – 4d = 201 3a + b = -40

  17. §9.3: Polynomial Models How do you find a polynomial model for a given set of data? Given the points: (1,1), (2,4), (3,10), (4,20) Find the degree of a function that would best model the function.

  18. The Tool: Polynomial Difference Theorem: The function y = f(x) is a polynomial function of degree n iff the nth difference of corresponding y-values are equal and non-zero.

  19. Example F(x) = 5x + 3 Pick x’s to plug in: x -2 -1 0 1 2 3 4 y’s -7 -2 3 8 13 18 23 1st diff: 5 5 5 5 5 5 This function is of degree 1.

  20. Example: X’s -1 0 1 2 3 4 5 f(x)=x2+x+3 3 3 5 9 15 23 33 1st difference 0 2 4 6 8 10 2nd difference 2 2 2 2 2 This is a 2nd degree model.

  21. Ex. Given the pairs, determine what the degree of the function is: X’s 0 1 2 3 4 5 6 Y’s -10 -7 8 47 122 245 428 1st diff: 3 15 39 75 123 183 2nd diff: 12 24 36 48 60 3rd diff: 12 12 12 12 Thus, a cubic model is best for the data. Use stat, Calc, cubic reg. to determine the fn. The function should be: f(x)= 2x3 + x - 10

  22. Determine if y is a polynomial function of x of degree less than 5.IF so, find an equation of least degree for y in terms of x.

  23. Assignment Section 9.3 P.547-576 #1,2,5-7, 9a-c 11a-c 15,16,20,21

  24. Warm-up 2/4/08 Let f(x) = 3x2 – 40x + 48 • Which of the given polynomials is a factor of f(x)? • x – 2 d) x - 6 • x – 3 e) x – 12 • x – 4 f) x – 24 • Which of the given values equals 0? • f(2) c) f(4) e) f(12) • f(3) d) f(6) f) f(24)

  25. Synthetic Division (dividing polynomials) When you are dividing by a linear factor, you can use a simplified process (synthetic division). Linear factor – graphed, it would be a line. In synthetic division -omit all variables & exponents -reverse the sign of the divisor so you can add

  26. Steps: • Reverse the sign of the divisor. Use 0 as a “place holder” for any missing term. • Bring down first coefficient. • Multiply the first coefficient by the divisor & write under next coefficient. • Add numbers, bring down. • Repeat steps 1-4 until done.

  27. Examples (Synthetic Division) • Divide x2 + 3x – 12 by x – 2 x + 5 – 2/x – 2 2) Divide x2 + 5x + 6 by x + 2 x + 3 3) Divide 2x2 – 19x + 24 by x – 8. 2x - 3

  28. 9.5: Polynomials & Linear Factors Sometimes its more useful to work with polynomials in factored form: Ex. X3 + 6x2 + 11x + 6 = (x + 1)(x + 2)(x + 3) How could you prove these are factors of the polynomial?

  29. Factor Theorem The expression (x – a) is a linear factor of a polynomial if and only if the value “a” is a zero of the related polynomial function. Ex. Graph y = x2 + 3x – 4 It crosses the x-axis at 1. Thus, (x – 1) is a factor of the polynomial. Also (x + 4) is a factor of the polynomial.

  30. If a function is in factored form, you can use the zero product property to find the values that will make the function equal zero. Ex. F(x) = (x – 1)(x + 2)(x – 4) For what values of x will f(x) = 0? 1, -2, 4

  31. What the answer tells you: In the previous example, (x – 1) and (x + 4) were the factors of the polynomial. Therefore: 1) -4 is a solution of x2 + 3x – 4 2) -4 is an x-intercept of y = x2 + 3x – 4 3) -4 is a zero of y = x2 + 3x – 4 4) x + 4 is a factor of y = x2 + 3x - 4

  32. Factor-Solution-Intercept Equivalence Theorem All the following are equivalent statements • (x – c) is a factor of f • f(c) = 0 • C is an x-intercept of the graph y = f(x) • C is a zero of f • The remainder when f(x) is divided by (x – c) is 0.

  33. Ex) Determine the zeros of the function y = (x – 2)(x + 1)(x + 3) x = 2, -1, -3 Ex) Write four equivalent statements about one of the solutions of the equation x2 – 4x + 3 = 0 Factors (x – 3)(x – 4)

  34. Ex1 p.585 Factor g(x) = 6x3 – 25x2 – 31x + 30 • Use a calculator/graph to see if any roots are obvious. • Verify the root (calculator or plug in) • Use the conjugate as a factor & complete long division • Continue factoring all terms of higher power than 1

  35. Finding Polynomials with Known 0’s Find an equation for a polynomial with zeros of -1, 4/5, and -8/3. Easy: (x + 1)(x – 4/5)(x + 8/3) = f(x) But, this is just one, to allow for other possibilities: If k is any non-zero term, this is more general: k(x + 1)(x – 4/5)(x + 8/3) = g(x)

  36. Assignment Section 9.5 P.587-588 #2-13, 17-18,20

  37. Warm-up 2/5/08 Find a value of k so that the graph of P(x) = 2x4 – 5x2 + k Intersects the x-axis in the indicated number of points. • 1 • 4 • 3 • 0

  38. Preferences Permanent Seats Want to move?

  39. §9.6: Complex Numbers Refresh Calculate (2 + 2i)4 Theorem If “a” and “b” are real numbers then a2 + b2 = (a + bi)(a – bi)

  40. Factor into linear factors: • 3p2 – 4 (√3p – 2 )(√3p + 2) 2) 4p2 – 9 (2p – 3)(2p + 3) 3) 4p2 + 9 (2p + 3i)(2p – 3i)

  41. Word Problem The volume of a container is modeled by the function V(x) = x3 – 3x2 – 4x. Let x represent the width, x + 1 the length, and x – 4 the height. If the container has a volume of 70 ft3, find its dimensions. 70 = x3 – 3x2 – 4x

  42. §9.7: The Fundamental Theorem of Algebra LEQ: How is the Fundamental Theorem of Algebra used to factor polynomials? Introduction… In textbook Read p.596-597 (through FToA)

  43. Fundamental Theorem of Algebra A polynomial function of degree “n” has exactly n zeros. -Some of the zeros may be imaginary and some may be multiple zeros. -You may have to use various methods to find all the answers: Methods: -graphing, factor theorem, polynomial division, and quadratic formula

  44. Zeros • IF “P” is a polynomial function of a degree n (greater than 1) with real coefficients, the graph can cross ANY horizontal line at most n times. • Ex) How many zeros does g(x) have? g(x) = -3x5 – xi

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