Unit 5POLYNOMIAL FUNCTIONS

Unit Essential Question: How are the properties of exponents and factoring going to help us solve polynomial functions?

Lesson 5.1PROPERTIES OF EXPONENTS

Lesson Essential Question: What are the different properties of exponents and how are they used?

Properties of Exponents: Product of Powers

Power of Powers

Power of Product

Negative Exponent

Zero Exponent

Quotient of Powers

Power of Quotient

Simplifying Expressions using Properties:

Scientific Notation: Must be in the form where c is and n is an integer.

Homework: Pages 333 – 334 #’s 3 – 39 odds, 43 – 45

Bell Work: Simplify each expression: 1)

2)

3)

Classwork/Homework: Page 333 #’s 4 – 42 evens and 46

This assignment is due by the end of class!!!

Bell Work: Simplify each expression:

Lesson 5.3SIMPLIFYING POLYNOMIAL EXPRESSIONS

Lesson Essential Question: How do we simplify polynomial expressions?

Adding and Subtracting Polynomials

Multiplying Polynomials

Homework: Pages 349 - 351 3 – 51 odds and 59 – 62

Bell Work: Simplify:

Lesson 5.4FACTORING AND SOLVING POLYNOMIAL FUNCTIONS

Lesson Essential Question: Why is it necessary to factor a polynomial expression in order to solve the equation?

GCF Factor the following expressions as much as possible: 1)

2)

3)

4)

Trinomials Factor each expression completely:

5)

6)

7)

8)

Factoring by Grouping Factor each expression completely: 9)

10)

Special Factoring Patterns Sum of Two Cubes:

Difference of Two Cubes:

Solving Polynomial Equations: Find all solutions for the following equations:

Homework: Pages 356 – 358 #’s 3 – 55 every other odd (3, 7, 11…), then 61 and 62

Bell Work: Find all solutions to the following equations:

1)

2)

3)

4)

Class Work: Pages 356 – 358 #’s 8, 10, 12, 16, 20, 24, 28, 32, 36, 38, 40, 44, 50, 52, 58, 60

This assignment will be collected tomorrow!

Bell Work: Find all of the solutions for the following equations: 1)

2)

Quiz Tomorrow: Properties of Exponents

Scientific Notation

Factoring Polynomial Expressions

Solving Polynomial Functions

Bell Work: Simplify each expression: 1)

2)

3) Find all solutions to the following equation:

Bell Work Continued: 4) Factor the following expression completely:

5) Find all solutions to the following equation:

6) The length, width, and height of a rectangular prism are x, x – 6, and x – 2. If the volume is 96 cubic feet, find the value of x.

Bell Work: Is (x+5) a factor of

Is (x – 2) a factor of

Lesson 5.5DIVISION OF POLYNOMIALS

Lesson Essential Question: What are the two ways we can divide polynomials and how do we use the remainder/factor theorem???

Long Division of Polynomials If and , divide g(x) by f(x).

If and , divide g(x) by f(x).

If and , divide g(x) by f(x).

Remainders!!! It is possible to get a remainder when you divide two polynomials.

Ex: If and , divide g(x) by f(x).

Ex: If and , divide g(x) by f(x).

Homework: Page 366 #’s 3 – 10

Bell Work: For 1 and 2, divide f(x) by g(x):

1)

2)

Example: Divide (x – 3) into

Synthetic Division: If you are dividing by a binomial in the form (x – c), then you can use synthetic division!!!

Examples:

Divide (x + 2) into

Divide (x -5) into

Divide (x + 4) into

Remainders!!! Yes you can still get remainders when you divide using synthetic division!!!

Ex: If and , divide g(x) by f(x).

Ex: If and , divide g(x) by f(x).

Homework: Page 366 #’s 11 – 18

Bell Work: For number 1, divide using long division. 1) divided by x + 3

For number 2, divide using synthetic division. 2) divided by x + 3

Remainder Theorem: When using synthetic division, the remainder that you get when you divide f(x) by some binomial (x – c) is the same value that you would get if you evaluated f(c).

This is easier shown by examples.

Factor Theorem: If the remainder you get when you use synthetic or long division is zero, this means that the original divisor you used is an actual factor for the polynomial.

Why does this help us???

This problem is CRAZY GOOD… If you know that (x – 3) is a factor of the polynomial , then rewrite f(x) as a product of all of its factors, then find all of the zeros for the function.

Example: If you know that (x + 2) and (x – 2) are factors of the polynomial , then rewrite f(x) as a product of all of its factors, then find all of the zeros for the function.

Example: If you know that (x – 2) is a factor of the polynomial , then rewrite f(x) as a product of all of its factors, then find all of the zeros for the function.

Homework: Pages 366 - 367 #’s 21 – 39 odds, and two word problems 41 and 42!!!

This assignment will be collected!

Bell Work: Be ready to go over last night’s homework!!!

Bell Work: What if we were not given a factor or zero for a given function? How could we break down a polynomial without knowing any other prior information about it?

Ex: Rewrite f(x) as a product of its factors and find the zeros for the function.

Lesson 5.6FINDING RATIONAL ZEROS

Lesson Essential Question: How do we find all possible rational zeros for a function if it will not factor and we do not know any prior information about the function?

Rational Zero Theorem: When dealing with a large polynomial function, we cannot always factor it easily. We can determine possible zeros by looking at the following:

Or… Use your graphing calculator to estimate where a zero might be, then test it using the factor theorem!!!

Examples: Express the following functions as a product of its factors, and find all zeros for the function. Then, sketch the graph of the function and find the intervals at which f(x) > 0 and f(x) < 0:

Bell Work: For the following functions, rewrite the function as a product of its factors, find all of the zeros, and then determine the intervals at which f(x) > 0 and f(x) < 0.

1)

2)

3)

Homework: Pages 374 - 376 #’s 11 – 17 odds, 25, 29, 33, 45

Find ALL ZEROS for these (this is includes complex and irrational zeros)!!!

For 11 – 17 odds, sketch the graph and find the intervals at which f(x) > 0 and f(x) < 0.

Bell Work: A rectangular prism has the following dimensions: (x – 3), (x + 5), and (x – 2). If the volume of the prism is 390 cubic inches, find the actual dimensions of the prism.

For , rewrite f(x) as a product of its factors, find its zeros, and find the intervals at which f(x) > 0 and f(x) < 0.

Classwork/Homework: Pages 374 - 375 #’s 12 – 18 evens and 28 – 34 evens 46, and 51

Find ALL ZEROS for these (this is includes complex and irrational zeros)!!!

For 12 – 18, find the intervals at which f(x) < 0 and f(x) > 0.

This will be collected!!!

Bell Work: Please get out your work from last night and be ready to ask questions.

Pop Quiz: For the following functions, rewrite f(x) as a product of its factors, find all zeros, and the intervals at which f(x) > 0 and f(x) < 0.

Test Upcoming!!! Test will be on: Factoring Long Division Synthetic Division Remainder/Factor Theorem Rewriting Polynomials in Factored Form Finding ALL Zeros for a Polynomial Function

Review: Factor each expression completely: 1) 2)

Find all solutions to the equations: 3) 4)

Review: Divide f(x) by g(x) using long division: 5) 6)

Divide f(x) by g(x) using synthetic division: 7) 8) 9) For 7-8, how can we check and see if we found the right remainder?

Review: Rewrite the following polynomials as a product of their factors, find the zeros, and the intervals at which f(x) > 0 and f(x) < 0.

10)

11)

12)

Review: 13) The volume of rectangular prism is given as . If the length is (x – 3) and the width is (x + 5), find the expression for the height.

14) A rectangular prism has a length and width of (x + 3) and height of (x² - 2). If the volume of the prism is 252 cubic feet, find the actual dimensions of the prism.

Bell Work: You have been charged with building the base of a statue that will consist of three rectangular blocks. The three blocks have the same height of x, and the base of each block is to be a square. The bottom block will have sides of (8x + 16), and each block above it will be half of the previous. The volume of the three cement blocks is to be 67,200 cubic inches. Find the dimensions of each block.

Examples: A ramp has a width that is twice its height, and a length that is 20 more than twice the width. If it took 576 cubic feet of concrete to make the ramp, find the dimensions.

Factor completely and find the zeros: