Properties of Exponents and Operations with Polynomials

1. Chapter 6 Polynomial Functions and Inequalities

2. 6.1 Properties ofExponents Negative Exponents a-n = • Move the base with the negative exponent to the other part of the fraction to make it positive

3. Product of Powers • am· an = *Add exponents when you have multiplication with the same base Quotient of Powers • = am – n *subtract exponents when you have division with the same base am+n

4. Power of a Power • (am)n = *Multiply exponents when you have a power to a power Power of a Product (ab)m= *Distribute the exponents when you have a multiplication problem to a power amn ambm

5. Power of a Quotient * distribute the exponent to both numerator and denominator, then use other property rules to simplify Zero Power a0 = * any number with the exponent zero = 1 1

6. Examples • 1. 52 ∙ 56 • 2. -3y ∙ -9y4 • 3. (4x3y2)(-5y3x) • 4. 597,6520

7. 5. (29m)0 + 70 6. 54a7b10c15 -18a2b6c5 • (3yz5)3 8. 8r4 2r -4

8. 9. y-7y 13. 30 – 3t0 y8 10. 4x0 + 5 14. (2-2 x 4-1)3 11. x-7y-2 x2y2 12. (6xy2)-1

9. 6.2 Operations with Polynomials • Polynomial: A monomial or a sum of monomials • Remember a monomial is a number, a variable, or the product of a number and one or more variables

10. Rules for polynomials • Degree of a polynomial = the degree of the monomial with the highest degree. *Remember, the degree of a monomial is the sum of the exponents of all the variables in the monomial. • Adding and Subtracting = combine like terms • Multiplying = Distribute or FOIL

11. Examples 1. (3x2-x+2) + (x2+4x-9) 2. (9r2+6r+16) – (8r2+7r+10) 3. (p+6)(p-9)

12. 4. 4a(3a2+b) 5. (3b-c)2 6. (x2+xy+y2)(x-y)

13. 6.3 Dividing Polynomials When dividing by a monomial: Divide each term by the denominator separately 1. 2.

14. Dividing by a polynomial • Long Division:rewrite it as a long division problem 1. 2.

15. Dividing by a polynomial • Synthetic Division • Step 1: Write the terms of the dividend so that the degrees of the terms are in descending order. Then write just the coefficients. • Step 2: Write the constant r of the divisor x – r to the left. Bring the first coefficient down. • Step 3: Multiply the first coefficient by r . Write the product under the second coefficient. Then add the product and the second coefficient. • Step 4: Multiply the sum by r. Write the product under the next coefficient and add. Repeat until finished. • Step 5: rewrite the coefficient answers with appropriate x values

16. Step 1 Steps 2-4 Step 5 1 - 4 6 - 4 1 ex 1: Use synthetic division to find (x3 – 4x2 + 6x – 4) ÷ (x – 2). x3 – 4x2 + 6x – 4 2 - 4 4  1 - 4 6 - 4 - 2 2 0 x2 – 2x + 2

17. Use synthetic division to solve each problem 2. 3.

18. 6.4 Polynomial Functions • Polynomial in one variable : A polynomial with only one variable • Leading coefficient: the coefficient of the term with the highest degree in a polynomial in one variable • Polynomial Function: A polynomial equation where the y is replaced by f(x)

19. State the degree and leading cofficient of each polynomial, if it is not a polynomial in one variable explain why. 1. 7x4 + 5x2 + x – 9 2. 8x2 + 3xy – 2y2 3. 7x6 – 4x3 + x-1 4. ½ x2 + 2x3 – x5

20. Evaluating Functions • Evaluate f(x) = 3x2 – 3x +1 when x = 3 • Find f(b2) if f(x) = 2x2 + 3x – 1 • Find 2g(c+2) + 3g(2c) if g(x) = x2 - 4

21. End Behavior • Describes the behavior of the graph f(x) as x approaches positive infinity or negative infinity. • Symbol for infinity

22. End behavior Practice f(x) - as x + f(x) - as x -

23. End Behavior Practice f(x) + as x + f(x) - as x -

24. End Behavior Practice f(x) + as x + f(x) + as x -

25. The Rules in General

26. To determine if a function is even or odd • Even functions: arrows go the same direction • Odd functions: arrows go opposite directions To determine if the leading coefficient is positive or negative • If the graph goes down to the right the leading coefficient is negative • If the graph goes up to the right then the leading coeffiecient is positive

27. The number of zeros • zeros are the same as roots: where the graph crosses the x-axis • The number of zeros of a function can be equal to the exponent or can be less than that by a multiple of 2. • Example a quintic function, exponent 5, can have 5, 3 or 1 zeros • To find the zeros you factor the polynomial Critical Points • points where the graph changes direction. • These points give us maximum and minimum values • Relative Max/Min

28. Put it all together • For the graph given • Describe the end behavior • Determine whether it is an even or an odd degree • Determine if the leading coefficient is positive or negative • State the number of zeros

29. Cont… • For the graph given • Describe the end behavior • Determine whether it is an even or an odd degree • Determine if the leading coefficient is positive or negative • State the number of zeros

30. For the graph given • Describe the end behavior • Determine whether it is an even or an odd degree • Determine if the leading coefficient is positive or negative • State the number of zeros

31. For the graph given • Describe the end behavior • Determine whether it is an even or an odd degree • Determine if the leading coefficient is positive or negative • State the number of zeros

32. 6.5/6.8 Analyze Graphs of Polynomial Functions • Descartes’ Rule: used to determine how many possible positive real zeros, negative real zeros or imaginary zeros a polynomial may have. • Total number of zeros = degree • In f(x) the number of sign changes is the same as number of positive real zeros (or less by a multiple of 2) • In f(-x) the number of sign changes is the same as the number of negative real zeros (or less by a multiple of 2) • Imaginary zeros are based on how many possible real zeros compared to the total number of zeros

33. Use Descartes’ rule to determine how many of each type of zero is possible. Ex 1. f(x) = x3 – x2 – 4x + 4

34. Find the location of all possible real zeros. Then name the relative minima and maxima as well as where they occur Ex 2. f(x) = x4 – 7x2 + x + 5

35. 6.9 Rational Zero Theorem Parts of a polynomial function f(x) • Factors of the leading coefficient = q • Factors of the constant = p • Possible rational roots =

36. Ex 1: List all the possible rational zeros for the given function a. f(x) = 2x3 – 11x2 + 12x + 9 b. f(x) = x3 - 9x2 – x +105

37. Relative Maximum and Minimum: the y-coordinate values at each turning point in the graph of a polynomial. *These are the highest and lowest points in the near by area of the graph At most each polynomial has one less turning point than the degree

38. Finding Zeros of a function • Use Descartes’ rule to find the number of possible zeros of each type • Find all the possible rational zeros then use synthetic division to find a number that gives you a remainder of 0 • Then factor and or use the quadratic formula with the remaining polynomial to find any other possible zeros • If the degree is 4 or higher- continue synthethic division until you have only x2…. remaining then do quadratic formula • If the degree is 3- do quadratic formula right away after finding the first zero

39. Find all the zeros of the given function • f(x) = 2x4 - 5x3 + 20x2 - 45x + 18

40. Ex 2: Find all the zeros of the given function f(x) = 9x4 + 5x2 - 4

41. Graphing Polynomials a. Rational Zero Theorem : find all zeros b: Make a table with the integers between the zeros c. plot all zeros and table ordered pairs d. Identify any relative minima or maxima

42. Graph: f(x) = x4 - 13x2 + 36

43. Graph: g(x) = 6x3 + 5x2 – 9x + 2

44. Graph: f(x) = x4 – x3 – x2 – x – 2