Chapter 2 – Polynomial, Power, and Rational Functions

1. Chapter 2 – Polynomial, Power, and Rational Functions HW: Pg. 175 #7-16

2. 2.1- Linear and Quadratic Functions and Modeling • Polynomial Functions- • Let n be a nonnegative integer and let a0, a1, a2,…, an-1, an be real numbers with an≠0. The functions given by f(x)=anxn + an-1xn-1+…+a2x2+a1x+a0 Is a polynomial function of degree n. The leading coefficient is an. f(x)=0 is a polynomial function. *it has no degree or leading coefficient.

3. Identify degree and leading coefficient for functions: • F(x) = 5x3-2x-3/4 • G(x) = √(25x4+4x2) • H(x) = 4x-5+6x • K(x)=4x3+7x7

4. Polynomial Functions of No and Low Degree

5. Linear Functions • F(x) = ax+b • Slope-Intercept form of a line: • Find an equation for the linear function f such that f(-2) = 5 and f(3) = 6

6. Average Rate of Change • The average rate of change of a function y=f(x) between x=a and x=b, a≠b, is [F(b)-F(a)]/[b-a]

7. Modeling Depreciation with a Linear Function Weehawken High School bought a $50,000 building and for tax purposes are depreciating it $2000 per year over a 25-yr period using straight-line depreciation. • What is the rate of change of the value of the building? • Write an equation for the value v(t) of the building as a linear function of the time t since the building was placed in services. • Evaluate v(0) and v(16) • Solve v(t)=39,000

8. Characteristics of Linear Functionsy=mx+b

9. Quadratic Functions and their graphs: Sketch how to transform f(x)=x2 into: • G(x)=-(1/2)x2+3 • H(x)=3(x+2) 2-1 • If g(x) and h(x) and in the form f(x)=ax2+bx+c, what do you notice about g(x) and h(x) when a is a certain value (negative or positive)?

10. Finding the Vertex of a Quadratic Function: • f(x)=ax2+bx+c We want to find the axis of symmetry, which is x=-b/(2a). Then: The graph of f is a parabola with vertex (x,y), where x=-b/(2a). If a>0, the parabola opens upward, and if a<0, it opens downward.

11. Use the vertex form of a quadratic function to find the vertex and axis of the graph of f(x)=8x+4x2+1: • x=-b/(2a)

12. Find the vertex of the following functions: • F(x)=3x2+5x-4 • G(x)=4x2+12x+4 • H(x)=6x2+9x+3 • f(x)=5x2+10x+5

13. Vertex Form of a Quadratic Function: • Any Quadratic Function f(x)=ax2+bx+c, can be written in the vertex form: • F(x)=a(x-h)2+k • Where (h,k) is your vertex • h=-b/(2a) and k=is the y

14. Using Algebra to describe the graph of quadratic functions: • F(x)=3x2+12x+11 f(x)=a(x-h)2+k =3(x2+4x)+11 Factor 3 from the x term =3(x2+4x+() - () )+11 Prepare to complete the square. =3(x2+4x+(2)2-(2)2)+11 Complete the square. =3(x2+4x+4)-3(4)+11 Distribute the 3. =3(x+2)2-1

15. Find vertex and axis, then rewrite functions in vertex form: f(x)=a(x-h)2+k • F(x)=3x2+5x-4 • F(x)=8x-x2+3 • G(x)=5x2+4-6x

16. Characteristics of Quadratic Functions: y=ax2+bx+c

17. 2.2 Power Functions With Modeling HW: Pg.189 #1-10

18. Power function • F(x)=k*xa • a is the power, k is the constant of variation EXAMPLES:

19. What is the power and constant of variation for the following functions: • F(x) = ∛x • 1/(x2) • What type of Polynomials are these functions? (HINT: count the terms)

20. Determine if the following functions are a power function Given that a,h,and c represent constants,, and for those that are, state the power and constant of variation: • 6cx-5 • h/x4 • 4∏r2 • 3*2x • ax • 7x8/9

21. 2.3 Polynomial Functions of Higher Degree HW: Pg 203 #33-42e

22. Graph combinations of monomials: • F(x)=x3+x • G(x)=x3-x • H(x)=x4-x2 • Find local extrema and zeros for each polynomial

23. Graph: F(x)=2x3 F(x)=-x3 F(x)=-2x4 F(x)=4x4 What do you notice about the limits of each function?

24. Finding the zeros of a polynomial function: • F(x)=x3—2x2-15x • What do these zeros tell us about our graph?

25. SKETCH GRAPHS: • F(x)=3x3 + 12x2 – 15x • H(x)=x2 + 3x2 – 16 • G(x)=9x3 - 3x2 – 2x • K(x)=2x3 - 8x2 + 8x • F(x)=6x2 + 18x – 24

26. 2.4 Real Zeros of Polynomial Functions HW: Pg. 216 #1-6

27. Long Division • 3587/32 (3x3+5x2+8x+7)/(3x+2)

28. Division Algorithm for Polynomials • F(x) = d(x)*q(x)+r(x) • F(x) and d(x) are polynomials where q(x) is the quotient and r(x) is the remainder

29. Fraction Form: F(x)/d(x)=q(x)+r(x)/d(x) • (3x3+5x2+8x+7)/(3x+2) • Write (2x4+3x3-2)/(2x2+x+1) in fraction form

30. Special Case: d(x)=x-k • D(x)=x-k, degree is 1, so the remainder is a real number • Divide f(x)=3x2+7x-20 by: • (a) x+2 (b) x-3 (c) x+5

31. We can find the remainder without doing long division! Remainder Theorem: If a polynomial f(x) is divided by x-k, then the remainder is r=f(k) Ex: (x2+3x+5)/(x-2) k=2 So, f(k)=f(2)=(2)2+3(2)+5=15=remainder

32. Lets test the Remainder Theorem with our previous example: • Divide f(x)=3x2+7x-20 by: • (a) x+2 (b) x-3 (c) x+5

33. PROVE: • If d(x)=x-k, where f(x)=(x-k)q(x) + r Then we can evaluate the polynomial f(x) at x=k:

34. Use the Remainder Theorem to find the remainder when f(x) is divided by x-k • F(x)=2x2-3x+1; k=2 • F(x)=2x3+3x2+4x-7; k=2 • F(x)=x3-x2+2x-1; k=-3

35. Synthetic Division • Now we can use this method to find both remainders and quotients for division by x-k, called synthetic division. • (2x3-3x2-5x-12)/(x-3) • K becomes zero of divisor

36. STEPS: * Since the leading coefficient of the dividend must be the leading coefficient , copy the first “2” into the first quotient position. * Multiply the zero of the divisor (3) by the most recent coefficient of the quotient (2). Write the product above the line under the next term (-3). * Add the next coefficient of the dividend to the product just found and record sum below the line in the same column. * Repeat the “multiply” and “add” steps until the last row is completed. • 3 | 2 -3 -5 -12 _____________

37. Use synthetic division to solve: • (x3-5x2+3x-2)/(x+1) • (9x3+7x2-3x)/(x-10) • (5x4-3x+1)/(4-x)

38. Rational Zero Theorem • Suppose f is a polynomial function of degree n1 of the form f(x)=anxn+…+a0 with every coefficient an integer. If x=p/q is a rational zero of f, where p and q have no common integer factors other than 1, then • P is an integer factor of the constant coefficient a0, and • Q is an integer factor of the leading coefficient an. Example: Find rational zeros of f(x)=x3-3x2+1

39. Finding the rational zeros: • F(x)=3x3+4x2-5x-2 Potential Rational Zeros:

40. Find rational zeros: • F(x)=6x3-5x-1 • F(x)=2x3-x2-9x+9

41. Upper and Lower Bound Tests for Real Zeros • Let f be a polynomial function of degree n≥1 with a positive leading coefficient. Suppose f(x) is divided by x-k using synthetic division. • If k≥0 and every number in the last line is nonnegative (positive or zero), then k is an upper bound for the real zeros of f. • If k≤0 and the numbers in the last line are alternately nonnegative and nonpositive, then k is a lower bound for the real zeros of f.

42. Example: • Lets establish that all the real zeros of f(x)=2x4-7x3-8x2+14x+8 must lie in the interval [-2,5]

43. Now we want to find the real zeros of the polynomial function f(x)=2x4-7x3-8x2+14x+8

44. Steps to finding the real zeros of a polynomial function: • Establish bounds for real zeros • Find the real zeros of a polynomial functions by using the rational zeros theorem to find potential rational zeros • Use synthetic division to see which potential rational zeros are a real zero • Complete the factoring of f(x) by using synthetic division again or factor.

45. Find the real zeros of a polynomial function: • F(x)=10x5-3x2+x-6

46. Find the real zeros of a polynomial function: • F(x)=2x3-3x2-4x+6 • F(x)=x3+x2-8x-6 • F(x)=x4-3x3-6x2+6x+8 • F(x)=2x4-7x3-2x2-7x-4

47. 2.5 Complex Numbers

48. F(x)=x2+1 has no real zeros • In the 17th century, mathematicians extended the definition of √(a) to include negative real numbers a. • i=√(-1) is defined as a solution of (i)2 +1=0 • For any negative real number √(a) = √|a|*i

49. Complex Number- is any number written in the form: • a +bi , where a, b are real numbers • a+bi is in standard form

50. Sum and Difference • Sum: (a+bi) +(c+di) = (a+c) + (b+d)i • Difference: (a+bi) – (c+di) = (a-c) + (b-d)I • EX: (a) (8 - 2i) + (5 + 4i) (b) (4 – i) – (5 + 2i)