1 Note-Taking Guides How to use these documents for success Print all the pages for the module. Open the first lesson on the computer. Fill in the guide as you read. Do the practice problems on notebook paper (usually). Put the notes and practice problems in a notebook. You can use these anytime! Review the notes before you go to sleep. Short term memory is converted to long term memory ONLY while you sleep. Your brain starts at the end of your day and converts things to long term memory in reverse order.
2 Module #6 6.01 Add Polys Vocabulary An expression that consists of one or more terms, including some variable(s). Examples: The highest degree of any term in the polynomial. Examples: Polynomial Degree A polynomial with one term (mono means one like monologue). Examples: A polynomial with two terms which are not like terms. (bi means two like bicycle). Examples: A polynomial with three terms which are not like terms (tri means three like tricycle). Examples: A polynomial with a degree of 1. Linear means line. These graphs can form a diagonal line.
3 Examples: A polynomial with a degree of 2. These graphs can form parabolas or U shaped graphs (in chapter 7). Examples: Finding the degree of a polynomial 3 Here the degree is. Recall that you need to to find the degree. In order to find the degree of a polynomial you need to find the degree of and choose the. What is the degree of each term? + so the degree is + so the degree is + so the degree is What is the degree of the polynomial? Addition of Polynomials Example 1 Since this problem is addition, you can do it Now (Hint: When there is no number in front of the variable, it is a 1.) This is the final answer. (Reorder from highest to lowest exponent. The 3 is actually.
4 Example 2 Since this problem is addition, you can do it Now Subtraction of Polynomials This is the final answer. Rule: Example 3: Since this is a subtraction problem, you must distribute the negative throughout the second parentheses. Notice that the signs changed in the second parentheses. Combine like terms. This is the final answer. Example 4: Since this is a subtraction problem, you must distribute the negative throughout the second parentheses. Notice that the signs changed in the second parentheses. Combine like terms. This is the final answer. Now, do the practice problems, check your answers, and seek help if you don t understand something.
5 6.02 Multiplication and Division of Polys In this lesson we will concentrate on Multiplication of monomials Example 1: x to the third means x to the fifth means (Write it out) You have a total of factors of x. Example 2: Multiply the (Write it out) Shortcut: Multiply the and add the. Copy example 3: Copy Example 4:
6 Division of monomials Example 1: There are of x in the and there are of x in the. Since is equal to 1, you can the common in both the. Cross out the common factors above. The answer is. Example 2: There are factors of and factor of in the numerator. There is factor of and factors of in the denominator. Cancel out the common factors including the numbers. The final answer is. Shortcut: Divide the and subtract the. Copy examples 3, 4, and 5 here. Now, do the practice problems, check your answers, and seek help if you don t understand something.
7 6.03 Powers Raising a power to a power How do you square a? Example 1: In the example below, the x squared has been raised to the power. This means. The rule of for multiplication is to, we have a total of or x to the 8 th power. Do you see any shortcuts? Example 2: In this example, is raised to the power or it means it is being multiplied times. Since there is a coefficient of -3, we will the -3. Write it out. Final answer Example 3: In this example, there is a coefficient of. Remember that 2 negatives is a. Write it out. Final answer Do you see a shortcut for the powers? If you're raising a power to a power, all you need to do is.
8 Putting it all together Some problems may require that you use both the power to a power rule and the product rule from 6.02. Step 1: In the first term, Now use the shortcut and multiply the In the second term, Now use the shortcut and multiply Step 2: To find the answer, you will need to multiply and then add. **Remember: If you are raising a power to a power, you the exponents. If you are multiplying, you the exponents. Now, do the practice problems, check your answers, and seek help if you don t understand something.
9 6.04 FOIL Multiplication of Polynomials (Monomial x Polynomial) Let's review the rules you will use in this lesson. When multiplying, coefficients are together and the exponents are. For example, is equal to. Multiply the and the to get -8. Since there are 3 factors of x in the first term and 6 factors of x in the second term, the answer will have. In this lesson, you will be multiplying. Let's begin with a monomial ( term) times a polynomial ( terms). Essentially, you are using the property and what you know about multiplying exponents. Step 1: Step 2: Step 3: Draw the arrows. Final answer: Multiplication of Polynomials (Binomial x Binomial) Let's use the problem (2x - 1) (x + 5) as an example. We must be sure to multiply every term in the polynomial with every term in the polynomial. Let's start with the 2x. (2x-1) (x+5) As shown in the illustration above, you must. You will multiply the 2x times the x and the 2x times the 5 to get. Now we will move on to the -1.
10 (2x-1) (x+5) As shown in the illustration above, you must. You will multiply the -1 times the x and the -1 times the 5 to get.to get the final answer, you must put it all together. There is an acronym to help you remember what to multiply. The acronym is FOIL. F stands for, O stands for, I stands for and L stands for. You will multiply the first, the terms on the, the terms on the and the last two terms. (2x-1) (x+5) Draw the arrows. Combine the like terms in the middle. Final answer Practice: (4x-3) (2x-7) (x+3) (x-3) (x+6y) (x-5y) Multiplication of Polynomials (Polynomial x Polynomial) Now you are ready for the big stuff. The last type of problem will involve the multiplication of many terms. Before we start, here are some important facts to remember. x times x is. Since you are multiplying, you must the exponents. x times is. Since you are multiplying, you must the exponents.
11 2x + 3x is. Since you are adding, only the will change, NOT the exponents. Terms must be to add them. and would not be like terms because the are not the same. We start by multiplying the x with all the terms in the parentheses. (x+3) ( Step 1: The next step is to multiply the 3 with all the terms in the second parentheses. (x+3) ( Step 2: Let's put it all together and collect the. Remember that when you are collecting like terms, you are. Only the of the like terms will be added. The exponents will NOT change. Final answer Here is one more example: Step 1: Step 2: Step 3: Now put all 3 steps together and combine the like terms. Remember to add the coefficients and NOT the exponents. Combine all 3 steps Final answer Now, do the practice problems, check your answers, and seek help if you don t understand something.
12 Lesson 6.06: Roots A square root is the opposite of In this lesson you will 1. Understand the associated with. Understand the. 2. Vocabulary Radical Symbol Read as Typed as Since x = 9 then Since x = 9 then = and also = Which one is the principle square root? Perfect Squares Roots that simplify into whole numbers are Complete the chart below. Since Then Since Then 1 X 1 = 1 1 = 1 6 x 6 = 36 36 = 6 3 x 3 = 9 9 = 3
13 Simplifying Square Roots The first thing that must be done is to factor the 18 into its prime factors. You may remember making a factor tree in an earlier math class. A factor tree is simply a method for "breaking down" a number to its prime factors. Copy the complete factorization of 18. Now we can write out the square root using the factorization of 18 and pull out the roots. **Remember the rule: Every 2 factors are equivalent to 1 factor. 18 So the square root of 18 is simplified to 3 times the square root of 2 and can be written 3 sqrt2 Step 1: Factor Tree Step 2: Write out the square root using the factorization of 72 Step 3: Pull out the roots. What is your answer?
14 Simplifying Square Roots with Variables Even exponents: Let's consider a variable with an exponent of 6. "a" to the sixth power means you have 6 factors of "a." Copy the rest of this example here. The final answer is because there are 3 factors of "a" outside the square root. There are no "leftovers" under the square root. Odd exponents: Let's consider a variable with an exponent of 9. "a" to the ninth power means you have 9 factors of "a." Copy the rest of this example here. The final answer is because there are 4 factors of "a" outside the square root and one "leftover" factor of "a" under the square root.
15 Simplifying Square Roots with Numbers and Variables Copy the rest of this example here. Copy the rest of this example here. Now, do the practice problems, check your answers, and seek help if you don t understand something.
16 Lesson 6.07: Multiplying and Dividing Roots To multiply radical expressions, Finally, you must Example 1 Multiply the numbers the square root and then multiply the numbers the square root. You get Now you have to the square root of 50. Remember every two factors under the square root is equal to one factor outside the square root. Copy the rest of example 1 here. Example 2 Copy example 2 here.
17 Example 3 Copy example 3 here. First multiply the numbers on the of the square root and then multiply the expressions on the of the square root. Don't forget that when you multiply you your exponents. The final answer is because there are 4 factors of "x" outside the square root and none "leftover" factor of "x" under the square root. Division of Radical Expressions The Steps 1. the fractions both on the inside and the outside of the square root: So here, we would simplify which is. Then, we would simplify which is. We would end up with 2. the square root To continue, we now have simplified our original expression to Keep simplifying. Our final answer will be
18 Rationalizing Radical Expressions If there is a in the denominator, you try to eliminate it by multiplying by. You have to get rid of the that is in the denominator. You will do this by multiplying the denominator by the square root of 2. The key here is that whatever you multiply in the, you must multiply in the So, in the problem shown above, you multiply by the Side Note: You are really multiplying by the number 1. You are not changing the of the answer, you are simply changing. Copy example 1 here. Finally, the square root of is equal to the whole number so the final answer is...
19 Practice Some Examples Example 1 Copy the steps for example 1. Step 1: Reduce Step 2: Simplify Example 2 Copy the steps for example 2. Include as much detail as you need. Example 3 Copy the steps for example 2. Include as much detail as you need. Now, do the practice problems, check your answers, and seek help if you don t understand something.
20 Lesson 6.09: Addition and Subtraction of Radicals Addition of Radicals Example 1: Step 1: to prime factor and take out a factor for every set of two of that factor. Step 2: Step 3: If any of the are the same then add/subtract their (numbers in front of them). Step 4: Do not change the! Copy example 1 here. Be sure you understand each step. Example 2 Copy example 2 here. Be sure you understand each step.
21 Addition and Subtraction of Radicals with Variables Example 1: Copy it here. Step 1: all square roots to and take out a factor for every set of two that factor. Step 2: Step 3: Put this in your own few words. Example 2: Step 1: all square roots to and take out a factor for every set of two that factor. Step 2: Step 3: Put this in your own few words. Now, do the practice problems, check your answers, and seek help if you don t understand something.
22 Lesson 6.10: Pythagorean Theorem Label the triangle below according to the diagram in the lesson. Here is the algebraic equation associated with the theorem. or Applying the Pythagorean Theorem Example 1 In the triangle below, use the Pythagorean Theorem to find the value of the missing side. Remember the theorem is A 2 + B 2 = C 2 where A and B are and is the hypotenuse. x is the of the triangle. 24 and 10 are the value of the sides. A 2 + B 2 = C 2 10 2 + 24 2 = x 2 In order to find the value of x, you must calculate the of both sides of the equation. The square root of 676 is and the square root of x 2 is. The length of the hypotenuse of the triangle is 26 cm.
23 Applying the Pythagorean Theorem Example 2 In the triangle below, use the Pythagorean Theorem to find the value of the missing side. The missing side is represented by x and is of the triangle. 3 is the value of the other side of the triangle and is the value of the hypotenuse. Start with A 2 + B 2 = C 2 and copy the steps here. Your Turn Practice Problems Use the Pythagorean Theorem to find the missing part of each right triangle shown below. Do your work right here. Check your answers from the link at the bottom of the activity. Number 1 Number 2 Number 3
24 Real-Life Examples Example 1 A 19-foot ladder just touches the bottom of a window. The bottom of the ladder is 9.25 feet from the base of the building. Which expression gives the height, in feet, from the ground to the bottom of the window? 1. 2. 3. 4. 5. Solution The Pythagorean Theorem is (side)2 + (side)2 = (hypotenuse)2. Remember that the hypotenuse is across from the right angle. Let's use x for the missing side. The given values will fit into the formula to give you the following equation : x 2 + 9.25 2 = 19 2 Our goal is (Here they subtracted 9.25 2 from both sides to get x 2 by itself.) Calculate the square root of both sides of the equation. So the answer is
25 Example 2 Use the diagram below to answer the question. Jim's mom will be staying with him for awhile. He wants to add a wheelchair ramp leading up to his back deck. How long will the planks be along the slope of the ramp? Give your answer to the nearest tenth. Solution Before we can use the Pythagorean Theorem, we must determine the measurements of both legs of the triangle formed by the ramp. Since every square on the grid is equal to 1 foot, the base of the triangle would measure feet. The height of the triangle would measure feet. The sides of the triangle are 6 ft and 2 ft so we can use the Pythagorean Theorem to find the hypotenuse of the triangle. Let x represent the hypotenuse or the slope of the ramp. Here is the equation. 6 2 + 2 2 = x 2 Keep copying this example. (Note: The directions say to round to the nearest tenth, but they simplified the square root instead. Feel free to just use a calculator in the last step if the directions say to round to the nearest tenth or something similar.) Now, do the practice problems, check your answers, and seek help if you don t understand something.