Chapter 6: Polynomials and Polynomial Functions. Chapter 6.1: Using Properties of Exponents Goals for this section. Properties of Exponents:

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1 Chapter 6: Polynomials and Polynomial Functions Chapter 6.1: Using Properties of Exponents Properties of Exponents: Products and Quotients of Powers activity on P323 Write answers in notes! Using Properties of Exponents Example: Evaluate each expression. (3 ) a b 5 8 ( y ) y y ( 2) ( 2) rs (rs )

2 Example: A circular component used in the manufacture of a microprocessor has a diameter of 200 mm and a thickness of 0.01 mm. What is the volume? Scientific Notation: Example: The red blood cells, white blood cells, and platelets found in human blood are all generated for the same stem cells. In laboratory experiments, scientists have found that as few as 10 stem cells can grow into 1,200,000,000,000 platelets in just four weeks. The number of white blood cells generated was the number of platelets. How many white blood cells were generated? Example: An average adult has about 528,000,000 ft of blood vessels in his or her body. How many times greater is this than the circumference of Earth, which is about 25,000 miles? Chapter 6.2: Evaluating and Graphing Polynomial Functions Polynomial function: In the form f(x) = a x + a x + + a x + a Leading coefficient: Constant term: Degree: Standard form:

3 Summary of Common Types of Polynomial Functions Degree Type Standard Form 0 Constant f(x) = a 1 Linear f(x) = a x + a 2 Quadratic f(x) = a x + a x + a 3 Cubic f(x) = a x + a x + a x + a 4 Quartic f(x) = a x + a x + a x + a x + a Example: Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type, and leading coefficient. f(x) = 2x x f(x) = 0.8x + x 5 Direct Substitution: Synthetic Substitution: Example: Use synthetic substitution to evaluate f(x) = 3x x 5x + 10 when x = -2. Example: Use synthetic substitution to evaluate f(x) = 5x + x 4x + 1 when x = 4. Example: To disarm a business security panel, you must push 4 buttons, no two of which have the same number or letter. The total number of ways to set the system is modeled by f(b) = b 6b + 11b 6b, where b is the number of buttons on the panel. Find how many ways the system can be set if there are 12 buttons on the panel.

4 Graphing Polynomial Function End behavior: Activity on Investigating End Behaviors P331 End Behavior for Polynomial Functions The graph of f(x) = a x + a x + + a x + a has this end behavior: For a > 0 and n even, f(x) + as x and f(x) + as x + For a > 0 and n odd, f(x) as x and f(x) + as x + For a < 0 and n even, f(x) as x and f(x) as x + For a < 0 and n odd, f(x) + as x and f(x) as x + Example: Graph f(x) = x + 2x x + 3 Example: The number of new words students in a language course were asked to learn each week is modeled by y = x + 50, where x is the number of weeks since the course began. Graph the model. Use the graph to estimate the number of words the students must learn in week 32.

5 Chapter 6.3: Adding, Subtracting, and Multiplying Polynomials Adding and Subtracting: Example: (5x + x 7) + (3x 6x 1) (3x + 8x x 5) (5x x + 17) (x + 2x + 8) + (2x 9) (9x 12x + x 8) (3x 12x x) Multiplying: Example: (4x + x 5)(2x + 1) (x + 2)(5x + 3x 1) (x 2)(x 1)(x + 3) Special Product Patterns:

6 Example: (3x 2)(3x + 2) (5a + 2) (2m 3) Example: From 1985 through 1996, the number of flu shots given in one city can be modeled by A = 11.33x 8.325x x 4190x for adults and by c = 6.87x + 106x 251x + 135x for children, where x is the number of years since Write a model for the total number T of flu shots given in these years. Example: From 1990 through 1996, the number of students S enrolled during the fall semester and the average number of credits C carried by each student can be modeled by s = x 1417x + 26,628 and C = 0.25x + 15, where x represents the number of years since Write a model for the total number of credits T carried by students x years after 1990.

7 Chapter 6.4: Factoring and Solving Polynomial Equations General Trinomial: Difference of Two Squares: Perfect Square Trinomial: GCF: Sum of Two Cubes Difference of Two Cubes Example: Factor each polynomial: x 64a 27a Factoring by Grouping: Example: Factor each polynomial: x y 3x 4y + 12 a b 8ab + 16b bx + 2a + 2b + ax x + 4x 21 25x 36

8 Example: Solve 2y 18y = 0 x + 7x 8x 56 = 0 Example: An optical company is going to make a glass prism that has a volume of 15 cm. the height will be h cm, and the base will be a right triangle with legs of length (h 2) cm and (h 3) cm. What will be the height? Chapter 6.5: The Remainder and Factor Theorems Dividing Polynomials: Example: Use long division to divide the polynomials. Divide y + 2y y + 5 by y y + 1 Divide 2x + 13x 7 by x + 6 Remainder Theorem: Example: Use synthetic division to divide x x 2x + 8 by each binomial. x 1 x 3x 7x + 6 by each binomial. x + 2 x + 2 x 4

9 Factor Theorem: Example: Factor f(x) = 3x + 13x + 2x 8 given that f( 4) = 0. Example: Factor f(x) = 3x + 14x 28x 24 given that f( 6) = 0. Example: One zero of f(x) = x + 6x + 3x 10 is x = 5. Find the other zeros of the function. Example: One zero of f(x) = 2x 9x 32x 21 is x = 7. Find the other zeros of the function. Example: A company that manufactures CD-ROM drives would like to increase its production. The demand function for the drives is p = 75 3x, where p is the price the company charges per unit when the company produces x million units. It costs the company $25 to produce each drive. Write an equation giving the company s profit as a function of the number of CD-ROM drives it manufactures. The company currently manufactures 2 million CD-ROM drives and makes a profit of $76,000,000. At what other level of production would the company also make $76,000,000?

10 Chapter 6.6: Finding Rational Zeros Using the Rational Zero Theorem: If f(x) = a x + a x + + a x + a has integer coefficients, then every rational zero of f has the following form: p q = factor of constant term a factor of leading coefficient a Example: Find the rational zeros of f(x) = x 4x 11x + 30 Example: Find the rational zeros of f(x) = x x 9x + 9 Example: Find all real zeros of f(x) = 15x 68x 7x + 24x 4 Example: Find all real zeros of f(x) = x 7x + 10x + 6 Example: A rectangular column of cement is to have a volume of ft. The base is to be square, with sides 3 ft less than half the height of the column. What should the dimensions of the column be?

11 Chapter 6.7: Using the Fundamental Theorem of Algebra The Fundamental Theorem of Algebra: Things to remember when solving polynomials: Example: State the number of solutions and tell what they are. x 14x + 49 = 0 x + 3x 8x 22x 24 Example: Find all the zeros of f(x) = x + x x + 15 Note: Previous example of f(x) = x + x x + 15 factors into If (x k) is raised to an odd power, the graph will cross the x-axis at x = k If (x k) is raised to an even power, the graph will be tangent to the x-axis at x = k which means it will be a repeat solution. All complex solutions come in complex conjugate pairs. If a + bi is a zero, then a bi is also a zero.

12 Example: Find all the zeros of f(x) = x + 5x 6. Example: Write a polynomial of least degree that has real coefficients, a leading coefficient of 1, and 1, 2 + i, and 2 i as zeros. Example: Write a polynomial function of least degree that has real coefficients, a leading coefficient of 1, and 5, 2i, and 2i as zeros. Using Technology to Approximate Zeros Example: Approximate the real zeros of f(x) = x 4x 5x Example: Approximate the real zeros of f(x) = x 3x 2x + 6. Example: A rectangular piece of sheet metal is 10 in. long and 10 in. wide. Squares of side length x are cut from the corner and the remaining piece is folded to make a box. The volume of the box is modeled by V(x) = 4x 40x + 100x. What size square can be cut form the corners to give a box with a volume of 25 in.

13 Chapter 6.8: Analyzing Graphs of Polynomial Functions Concept Summary: Let f(x) = a x + a x + + a x + a be a polynomial function. The following statements are equivalent: Zero: Factor: Solution: X-Intercept: Example: Graph f(x) = 2(x 9)(x + 4). Example: Graph f(x) = x(2x 5)(x + 5).

14 Turning Points: Number of Turning Points: Example: Graph each function. Identify the x-intercepts, local maximums, and local minimums. f(x) = x + 2x 5x + 1 f(x) = 2x 5x 4x 6 Example: You want to make a rectangular box that is x cm high, (x + 5) cm long, and (10 x) cm wide. What is the greatest volume possible? What will the dimensions of the box be?

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