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Chapter 2

Chapter 2. Polynomial and Rational Functions. Section 1. Quadratic Functions. Quadratic Functions. Let a, b , and c be real number with a ≠ 0 . The function f(x) = ax 2 + bx = c is called a quadratic function.

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Chapter 2

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  1. Chapter 2 Polynomial and Rational Functions

  2. Section 1 Quadratic Functions

  3. Quadratic Functions • Let a, b, and c be real number with a ≠ 0. The function f(x) = ax2 + bx = c is called a quadratic function. • The graph of a quadratic function is a special type of U-shaped curve that is called a parabola. • All parabolas are symmetric with respect to a line called the axis of symmetry, or simply the axis of the parabola. • The point where the axis intersects the parabola is called the vertex.

  4. Quadratic Functions • If a >0, then the graph opens upward. • If a < 0, then the graph opens downward.

  5. The Standard Form of a Quadratic Function The standard form of a quadratic functions • f(x) = a(x-h)2+ k, a ≠ 0 • Vertex is (h, k) • |a| produces a vertical stretch or shrink • (x – h)2represents a horizontal shift of h units • k represents a vertical shift of k units • Graph by finding the vertex and the x-intercepts

  6. Vertex of a Parabola The vertex of the graph f(x) = a(x)2+ bx + c is ( -b/2a, f(-b/2a)) EXAMPLE Find the vertex and x-intercepts -4x2 +x + 3

  7. Section 2 Polynomial Functions of Higher Degree

  8. Polynomial Functions Let n be a nonnegative integer and let an, an-1, …..…a2, a1, a0 be real numbers with an ≠ 0. The function f(x) = anxn + an-1xn-1 +…… a2x2 + a1x + a0 is called a polynomial function of x with degree n. EXAMPLE f(x) = x3

  9. Characteristics of Polynomial Functions • The graph is continuous. • The graph has only smooth rounded turns.

  10. Sketching Power Functions Polynomials with the simplest graphs are monomial of the form f(x) = xnand are referred to as power functions. REMEMBER ODD and EVEN FUNCTIONS • Even : f(-x) = f(x) and symmetric to y-axis • Odd: f(-x) = - f(x) and symmetric to origin

  11. Leading Coefficient Test • When n is odd: If the leading coefficient is positive (an >0), the graph falls to the left and rises to the right • When n is odd: If the leading coefficient is negative (an <0), the graph rises to the left and falls to the right

  12. Leading Coefficient Test • When n is even: If the leading coefficient is positive (an >0), the graph rises to the left and right. • When n is even: If the leading coefficient is negative (an <0), the graph falls to the left and right

  13. EXAMPLE • Identify the characteristics of the graphs • f(x) = -x3 + 4x • f(x) = -x4 - 5x2 + 4 • f(x) = x5 - x

  14. Real Zeros of Polynomial Functions If f is a polynomial function and a is a real number, the following statements are equivalent. • x = a is a zero of the function f • x =a is a solution of the polynomial equation f(x)=0 • (x-a) is a factor of the polynomial f(x) • (a,0) is an x-intercept of the graph of f

  15. Repeated Zeros A factor (x-a)k, k >0, yields a repeated zero x = a ofmultiplicity k. • If k is odd, the graph crosses the x-axis at x = a • If k is even,the graph touches the x-axis at x = a (it does not cross the x-axis)

  16. EXAMPLE • Graph using leading coefficient test, finding the zeros and using test intervals • f(x) = 3x4 -4x3 • f(x) = -2x3 + 6x2 – 4.5x • f(x) = x5 - x

  17. Section 3 Long Division of Polynomials

  18. Long Division Algorithm If f(x) and d(x) are polynomials such that d(x) ≠ 0, and the degree of d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x) and r(x) such that: f(x) = d(x) q(x) + r(x)

  19. EXAMPLE Divide the following using long division. • x3 -1 by x – 1 • 2x4 + 4x3 – 5x2 + 3x -2 by x2 +2x – 3 Remember to use zero coefficients for missing terms

  20. Synthetic Division Synthetic Division is simply a shortcut for long division, but you still need to use 0 for the coefficient of any missing terms. EXAMPLE Divide x4 – 10x2 – 2x +4 by x + 3 -3 1 0 -10 -2 4 -3 9 3 -3 1 -3 -1 1 1 = x3 – 3x2 - x + 1 R 1

  21. The Remainder and Factor Theorems Remainder Theorem: If a polynomial f(x) is divided by x-k, then the remainder is r = f(k) EXAMPLE Evaluate f(x) = 3x3 + 8x2 + 5x – 7 at x = -2 Using synthetic division you get r = -9, therefore, f(-2) = -9

  22. The Remainder and Factor Theorems Factor Theorem: A polynomial f(x) has a factor (x-k) if and only if f(k) =0 EXAMPLE Show that (x-2) and ( x+3) are factors of f(x) = 2x4 + 7x3 -4x2 -27x – 18 Using synthetic division with x-2 and then again with x+3 you get f(x) = (x-2)(x+3)(2x+3)(x+1) implying 4 real zeros

  23. Uses of the Remainder in Synthetic Division The remainder r, obtained in the synthetic division of f(x) by x-k, provides the following information: • The remainder r gives the value of f at x=k. That is, r= f(k) • If r=0, (x-k) is a factor of f(x) • If r=0, (k,0) is an x-intercept of the graph of f

  24. Section 4 Complex Numbers

  25. The Imaginary Unit i Because some quadratic equations have no real solutions, mathematicians created an expanded system of numbers using the imaginary unit i, defined as i = -1 i2 = -1 i3 = -i i4 = 1

  26. Complex Numbers The set of complex numbers is obtained by adding real numbers to real multiples of the imaginary unit. Each complex number can be written in the standard form a + bi . If b = 0, then a + bi = a is a real number. If b ≠ 0, the number a + bi is called an imaginary number. A number of the form bi, where b ≠ 0, is called a pure imaginary number.

  27. Some Properties of Complex Numbers • a + bi = c+ diif and only if a=c and b=d. • (a + bi) + (c+ di) = (a +c) + (b + d)i • (a + bi) – (c+ di) = (a – c) + (b – d)i • – (a + bi) = – a – bi • (a + bi ) + (– a – bi) = 0 + 0i = 0

  28. Complex Conjugates a + bi and a –bi are complex conjugates (a + bi) (a –bi ) = a2 + b2 EXAMPLE (4 – 3i) (4 + 3i) = 16 + 9 = 25

  29. Complex Solutions of Quadratic Equations Principal Square Root of a Negative Number If a is a positive number, the principal square root of the negative number –a is defined as  – a = a i EXAMPLE •  – 13 = 13i

  30. Section 5 TheFundamental Theorem of Algebra

  31. The Fundamental Theorem of Algebra Linear Factorization Theorem If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system. If f(x) is a polynomial of degree n, where n > 0, then f has precisely n linear factors. f(x) = an(x – c1)(x-c2)…(x–cn) Where c1,c2…cnare complex numbers

  32. EXAMPLE Find the zeros of the following: • f(x) = x – 2 • f(x) = x2 – 6x + 9 • f(x) = x3 + 4x • f(x) = x4 – 1

  33. Rational Zero Test If the polynomial f(x)= anxn + an-1xn-1 +…a2x2+a1x1 +a0has integer coefficients, every rational zero of f has the form Rational zero = p/q or constant term/leading coefficent Where p and q have no common factors other than 1, and p = a factor of the constant term a0 q = a factor of the leading coefficient an

  34. EXAMPLE Find the rational zeros of f(x) = 2x3+3x2 – 8x + 3 Rational zeros p/q = ± 1, ± 3 / ± 1, ± 2 Possible rational zeros are ± 1, ± 3, ± ½, ± 3/2 Use synthetic division by trial and error to find a zero

  35. Conjugate Pairs Let f(x) be a polynomial function that has real coefficients. If a + bi, where b ≠ 0, is a zero of the function, the conjugate a – bi is also a zero of the function. Rational zero = p/q Where p and q have no common factors other than 1, and p = a factor of the constant term a0 q = a factor of the leading coefficient an

  36. EXAMPLE Find a 4th degree polynomial function with real coefficients that has – 1, – 1, and 3i as zeros Then f(x) = a(x+1)(x+1)(x – 3i)(x+3i) For simplicity let a = 1 Multiply the factors to find the answer.

  37. EXAMPLE Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 where 1 + 3i is a zero Knowing complex zeros occur in pairs, then 1 – 3i is a zero Multiply (1+3i)(1 – 3i) = x2 – 2x +10 and use long division to find the other zeros of -2 and 3 x4 – 3x3 + 6x2 + 2x – 60/(x2 – 2x +10)

  38. EXAMPLE Find all the zeros of f(x) = x5 + x3 + 2x2 – 12x +8 Find possible rational roots and use synthetic division

  39. EXAMPLE You are designing candle-making kits. Each kit will contain 25 cubic inches of candle wax and a mold for making a pyramid-shaped candle. You want the height of the candle to be 2 inches less than the length of each side of the candle’s square base. What should the dimension of your candle mold be? Remember V = 1/3Bh

  40. Section 6 Rational Functions

  41. Rational Function A rational function can be written in the form f(x) = N(x)/D(x) where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. Also, this sections assumes N(x) and D(x) have no common factors. In general, the domain of a rational function of x includes all real numbers except x-values that make the denominator zero.

  42. EXAMPLE Find the domain of the following and explore the behavior of f near any excluded x-values (graph) • f(x) = 1/x • f(x) = 2/(x2 – 1) 2

  43. ASYMPTOTE Is essentially a line that a graph approaches but does not intersect.

  44. Horizontal and Vertical Asymptotes • The line x= a is a vertical asymptote of the graph of f if f(x) → ∞ or f(x) → – ∞ as x → a either from the right or from the left. • The line y= b is a horizontal asymptote of the graph of f if f(x) → b as x → ∞ or x → – ∞

  45. Asymptotes of a Rational Function Let f be the rational function given by f(x) = N(x)/D(x) where N(x) and D(x) have no common factors then: anxn + an-1xn-1…./(bmxm +bm-1xm-1…) • The graph of f has vertical asymptotes at the zeros of D(x). • The graph of f has one or no horizontal asymptote determined by comparing the degrees of N(x) and D(x) • If n < m, the graph of f has the line y = 0 as a horizontal asymptote. • If n= m, the graph of f has the line y = an/bmas a horizontal asymptote. • If n>m, the graph of f has no horizontal asymptote

  46. EXAMPLE Find the horizontal and vertical asymptotes of the graph of each rational function. • f(x) = 2x/(x4 + 2x2 + 1) • f(x) = 2x2 /(x2 – 1)

  47. Graphing Rational Functions Let f be the rational function given by f(x) = N(x)/D(x) where N(x) and D(x) have no common factors • Find and plot the y-intercept (if any) by evaluating f(0). • Find the zeros of the numerator (if any) by solving the equation N(x) =0 and plot the x-intercepts • Find the zeros of the denominator (if any) by solving the equation D(x) = 0, then sketch the vertical asymptotes • Find and sketch the horizontal asymptote (if any) using the rule for finding the horizontal asymptote of a rational function • Test for symmetry (mirror image) • Plot at least one pointbetweenand one point beyond each x-intercept and vertical asymptote • Use smooth cures to complete the graph between and beyond the vertical asymptotes

  48. EXAMPLE Graph • f(x) = 3/(x – 2) • f(x) = (2x – 1)/x • f(x) = (x2 – 9)/(x2 – 4)

  49. Slant Asymptotes Consider a rational function whose denominator is of degree 1 or greater. If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant (or oblique) asymptote.

  50. EXAMPLE f(x) = (x2 – x) /( x+ 1) has a slant asymptote. To find the equation of a slant asymptote, use long division. You get x – 2 + 2/(x+1) y = x – 2 because the remainder term approaches 0 as x increases or decreases without bound

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