1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The formal definition of a power function is given below: POWER FUNCTIONS An function of the form: where a and b are real numbers not equal to zero. Eercise #1: For each of the following power functions, state the value of a and b b writing the equation in the b form a. 3 (a) (b) 1 7 (c) 8 (d) 3 3 6 The characteristics of power functions depend on both the value of a and the value of b. The most important, though, is the eponent, b. Eercise #: Consider the general power function b a. (a) What can be said about the -intercept of an power function if b 0? Illustrate. (b) What can be said about the -intercept of an power function if b 0? Illustrate. For now we will just concentrate on power function where the eponent is a positive whole number. Eercise #3: Using our table, fill in the following values for common power functions. 3 1 0 1 3 3 4 5 emathinstruction, RED HOOK, NY 1571, 015
From the previous eercise, we should note that when the power function has an even eponent, then positive and negative INPUTS have the same value. When the power function has an odd eponent, then positive and negative inputs have opposite outputs. Eercise #4: Using our calculators, sketch the power functions below using the standard viewing window. (a) (b) 3 (c) 4 (d) 5 Eercise #5: Which of the following power functions is shown in the graph below? Eplain our choice. Do without the use of our calculator. (1) 7 4 (3) 6 8 () 10 3 (4) 5 9 The End Behavior of Polnomials The behavior of polnomials as the input variable gets ver large, both positive and negative, is important to understand. We will eplore this in the net eercise. 3 3 Eercise #6: Consider the two functions 1 9 30 and. (a) Graph these functions using min 10, ma 10, min 100, ma 100 (b) Graph these functions using min 0, ma 0, min 1000, ma 1000 (c) Graph these functions using min 50, ma 50, min 10000, ma 10000 (d) Graph these functions using min 100, ma 100, min 100000, ma 100000 (e) What do ou observe about the nature of the two graphs as the viewing window gets larger? (f) Wh is this occurring? The end behavior (also known as long-run) of an polnomial is dictated b its highest powered term!!! emathinstruction, RED HOOK, NY 1571, 015
3 Generalizations for Power Functions Sketch the general shape of a power function given each of the following criteria. Assume values of a are positive integers. where a is odd. a, where a is even. a, where a is odd a, where a is even. a, emathinstruction, RED HOOK, NY 1571, 015
4 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Without using our calculator, determine which of the following equations could represent the graph shown below. Eplain our choice. (1) () 3 (3) 4 (4) 5. Identif which of the following are power functions. For each that is a power function, write it in the form n a, where a and n are real numbers. Placing them in these forms ma take some mindful algebraic manipulation. 5 (a) 3 (b) 5 4 7 (c) 5 10 (d) 6 7 3 (e) 7 (f) 5 7 48 (g) 4 (f) 3 3. If the point (-3, 8) lies on the graph of a power function with an even eponent, which of the following points must also lie its graph? (1) 3, 8 (3) 3, 8 () 3, 8 (4) 8, 3 emathinstruction, RED HOOK, NY 1571, 015
5 4. If the point (-5, 1) lies on the graph of a power function with an odd eponent, which of the following points must also lie on its graph? (1) 5, 1 () 1, 5 (3) 5,1 (3) 1, 5 5. For each of the following polnomials, give a power function that best represents the end behavior of the polnomial. (a) 3 3 1 (b) 10 8 (c) 5 3 6 4 10 (d) 5 4 3 4 7 (e) 4 5 (f) 5 7 3 4 8 3 6 The graph below could be the long-run behavior for which of the following functions? Do this problem without graphing each of the following equations. (1) 7 1 3 () 4 6 4 4 3 (3) 5 3 9 5 (4) 3 4 1 REASONING 7. Let's eamine wh end-behavior works a little more closel. Consider the functions 3 f and 3 g 7 10. f (a) Fill out the table below for the values of listed. Round our final column to the nearest hundredth. (Do this on the homescreen with the STORE feature.) (b) What number is the ratio in the fourth column approaching as gets larger? What does this tell ou about the part of g that can be attributed to the cubic term? 5 10 50 100 f g g emathinstruction, RED HOOK, NY 1571, 015
6 LESSON #9 - SOLVING HIGHER ORDER EQUATIONS COMMON CORE ALGEBRA II Higher order polnomial equations are those where the variable is cubed, to the fourth power, or higher. An eample would be. Some of these equations, including the one in this paragraph, require advanced factoring techniques to solve. The ones we will be working with in this course can all be solved using the techniques ou alread know for quadratics including factoring, the quadratic formula, and incomplete quadratics. What is the maimum number of solutions for a linear equation? quadratic equation? cubic equation? How can ou generalize the maimum number of solutions to a polnomial equation? The following three situations summarize the different methods we will be using. 1. With a GCF Notes (we have alread done some like this) 5 4 3 5 0 15 Solve for zero if necessar. 5 4 3 5 0 15 0 3 5 ( 4 3) 0 3 5 ( 3)( 1) 0 3 5 0 or 0 or 3 0 or 1 0 5 0 or 0 or 3 or 1 3, 1,0 Factor out the GCF. Factor again if possible. Set each factor equal to zero and solve. If an of our new equations is a quadratic use the quadratic formula or the method for incomplete quadratics to solve that part of the equation rest of the equation.. In the form: 4 a b c 0 4 13 36 13 36 0 4 ( 9)( 4) 0 9 0 or 4 0 9 4 3 3,,,3 Notes Solve for zero if necessar. Notice that the equation is similar to 13 36 0 which is equal to ( 9)( 4) 0 in factored form. If we replace in each factor with, 4 13 36 ( 9)( 4). Set each factor equal to zero and solve. You can usuall use the method we learned for incomplete quadratics. Note: This method will work with trinomials in the form 4 a b c. emathinstruction, RED HOOK, NY 1571, 015
7 3. Factoring b Grouping Notes 3 5 15 6 0 This method is used with a cubic that does not 3 have a GCF. 5 15 6 0 5 ( 3) ( 3) 0 (5 )( 3) 0 5 0 or 3 0 5 or 3 5 10 5 5 5 5 Factor b grouping Set each factor equal to zero. Solve each resulting equation using an appropriate method. 10 10 3,, 5 5 Note: Solving polnomials of higher degree takes practice and some creativit to find the best method. As long as the solutions are not imaginar, ou could also use a graph to find the roots like we learned last unit. The onl problem is that irrational solutions will be estimated on the graph, not in radical form. Look at 1. Where would the graph of 5 4 3 5 0 15 intersect the -ais? Look at. Where would the graph of 4 13 36 intersect the -ais? Look at 3. Where would the graph of tenth. 3 5 15 6 intersect the -ais? Round to the nearest Solve each of the following equations. 5 3 1. t 10t 1t 0. 3 a a a 3 3 0 emathinstruction, RED HOOK, NY 1571, 015
8 3. 9 100 4. 4 5 3 5 60 160 0 5. 0 9 3 6. 5 3 6 8 0 7. 3 3 3 0 8. 4 1 9 0 9. 4 16 5 0 emathinstruction, RED HOOK, NY 1571, 015
9 Solve each of the following equations. LESSON #9 - SOLVING HIGHER ORDER EQUATIONS COMMON CORE ALGEBRA II HOMEWORK 1. 3 7 10 0. 3 3 4 1 0 3. 3 3 3 0 4. 5 4 0 4 5. 4 81 0 6. 4 10 9 0 emathinstruction, RED HOOK, NY 1571, 015
10 7. 3 18 0 8. 1 3 1 0 9. 5 4 3 4 0 10. 16 1 0 11. ( 1)( 3 4) 0 3 1. 6 3 0 emathinstruction, RED HOOK, NY 1571, 015
11 LESSON #30 - GRAPHS AND ZEROES OF A POLYNOMIAL COMMON CORE ALGEBRA II A polnomial is a function consisting of terms that all have whole number powers. In its most general form, a polnomial can be written as: n n 1 a a a a n n 1 1 0 Quadratic and linear functions are the simplest of all polnomials. In this lesson we will eplore cubic and quartic 3 4 functions, those whose highest powers are and respectivel. Eercise #1: For each of the following cubic functions, sketch the graph and circle its -intercepts. (a) 3 3 6 8 (b) 3 8 9 (c) 3 1 18 Clearl, a cubic ma have one, two or three real roots and can have two turning points. Just as with parabolas, there eists a tie between a cubic s factors and its -intercepts. Eercise #: Consider the cubic whose equation is (a) Determine the general shape of the function. (b) Determine the -intercept. 3 1. (e) Complete the sketch of this function. Check to be sure it has the general shape ou determined in part (a). (c) Algebraicall determine the zeroes of this function. Plot the zeroes on the grid to the right. (d) Use our graphing calculator to find the relative maimum and minimum points on the graph to the nearest tenth. emathinstruction, RED HOOK, NY 1571, 015
1 Eercise #3: The largest root of 3 (1) 4 and 5 (3) 10 and 11 () 6 and 7 (4) 8 and 9 9 1 0 falls between what two consecutive integers? Eercise #4: Consider the quartic function 4 5 4. (e) Complete the sketch of this function. Check to (a) Determine the general shape of the function. be sure it has the general shape ou determined in part (a). (b) Determine the -intercept. (c) Algebraicall determine the -intercepts of this function. (d) Use our graphing calculator to find the relative maimum and minimum points on the graph to the nearest tenth. Find an appropriate viewing window. emathinstruction, RED HOOK, NY 1571, 015
13 Eercise #5: Consider the quartic whose equation is (a) Determine the general shape of the function. (b) Determine the -intercept. 4 3 3 35 39 70. (d) What are the solutions to the equation, 4 3 3 35 39 70 0? (c) Sketch a graph of this quartic on the aes below. Label its -intercepts. Find an appropriate viewing window. (e) Write the epression in its factored form. 4 3 3 35 39 70 emathinstruction, RED HOOK, NY 1571, 015
14 FLUENCY LESSON #30 - GRAPHS AND ZEROES OF A POLYNOMIAL COMMON CORE ALGEBRA II HOMEWORK 1. Consider the cubic function 3 8. (a) Algebraicall determine the zeroes of this cubic function. (d) Sketch the function on the aes given. Clearl plot and label each -intercept. (b) Determine the -intercept. (c) Use our graphing calculator to find the relative maimum and minimum points on the graph to the nearest tenth. emathinstruction, RED HOOK, NY 1571, 015
15. Consider the cubic function 3 36 7. (a) Find an appropriate -window for the -window shown on the aes and sketch the graph. Make the sure the window is sufficientl large to show the two turning points and all intercepts. Clearl label all -intercepts. (b) What are the solutions to the equation 3 36 7 0? (c) Based on our answers to (b), how must the 3 epression 36 7 factor? 3. Consider the cubic function given b 3 6 1 5. (a) Sketch a graph of this function on the aes given below. Label the and intercepts. Round to the nearest tenth when necessar. (b) Considering the graphs of cubics ou saw in class and those in problems 1 and, what is different about the wa this cubic s graph looks compared to the others? emathinstruction, RED HOOK, NY 1571, 015
16 4. Consider the quartic function 4 3 7 5 50. (a) What is the general shape of the function? (d) Use our graph from part (c) to solve the 4 3 equation 7 5 50 0. (b) What is the -intercept of the function. (c) Sketch the graph of this function on the aes given below. Clearl mark all -intercepts, - intercepts, and relative etrema. Round to the nearest tenth when necessar. (e) Considering our answer to (d), how does the 4 3 epression 7 5 50 factor? 5. In general, how does the number of zeroes (or -intercepts) relate to the highest power of a polnomial? Be specific. Can ou make a statement about the minimum number of zeroes as well as the maimum? a. Maimum number of zeroes (alwas): b. Minimum number of zeros (when the highest power is even): c. Minimum number of zeros (when the highest power is odd): emathinstruction, RED HOOK, NY 1571, 015
17 LESSON #31 - CREATING POLYNOMIAL EQUATIONS COMMON CORE ALGEBRA II The connection between the zeroes of a polnomial and its factors should now be clear. This connection can be used to create equations of polnomials. The ke is utilizing the factored form of a polnomial. THE FACTORED FORM OF A POLYNOMIAL If the set represent the roots (zeroes) of a polnomial, then the polnomial can be written as: where a is some constant determined b another point Eercise #1: Determine the equation of a quadratic function whose roots are 3 and 4 and which passes through the point, 50. Epress our answer in standard form ( a b c ). Verif our answer b creating a sketch of the function on the aes below. It s important to understand how the a value effects the graph of the polnomial. This is easiest to eplore if the polnomial remains in factored form., where a 0. Eercise #: Consider quadratic polnomials of the form a 5 (a) What are the -intercepts of this parabola? (b) Sketch on the aes given the following equations: 5 5 4 5 emathinstruction, RED HOOK, NY 1571, 015
18 As we can see from this eercise, the value of a does not change the zeroes of the function, but does verticall stretch the function. We can create equations of higher powered polnomials in a similar fashion. Eercise #3: Create the equation of the cubic, in standard form, that has -intercepts given b the set 4,,5 and passes through the point 6, 0. Verif our answer b sketching the cubic s graph on the aes below. Eercise #4: Create the equation of a cubic in standard form that has a double zero at and another zero at 4. The cubic has a -intercept of 16. Sketch our cubic on the aes below to verif our result. Choose an appropriate viewing window. Eercise #5: How would ou describe this cubic curve at its double root? emathinstruction, RED HOOK, NY 1571, 015
19 FLUENCY LESSON #31 - CREATING POLYNOMIAL EQUATIONS COMMON CORE ALGEBRA II HOMEWORK 1. Create the equation of a quadratic polnomial, in standard form, that has zeroes of 5 and and which passes through the point 3, 4. Sketch the graph of the quadratic below to verif our result. Choose an appropriate viewing window.. Create an equation for a cubic function, in standard form, that has -intercepts given b the set 3,1, 7 and which passes through the point, 54. Sketch our result on the aes shown below. emathinstruction, RED HOOK, NY 1571, 015
0 3. Create the equation of a cubic whose -intercepts are given b the set 6, 3, 5 and which passes through the point 3, 36. Note that our leading coefficient in this case will be a non-integer. Sketch our result below. 4. Create the equation of a quadratic function, in standard form, that has one zero of 3 and a turning point at 1, 16. Hint tr to determine the second zero of the parabola b thinking about the relationship between the first zero and the turning point (ais of smmetr). Sketch our solution below. emathinstruction, RED HOOK, NY 1571, 015
1 LESSON #3 - POLYNOMIAL LONG DIVISION COMMON CORE ALGEBRA II We have worked to simplif rational epressions (polnomials divided b polnomials). In this lesson, we will look more closel at the division of two polnomials and how it is analogous to the division of two integers. Eercise #1: Consider the division problem 1519 7, which could also be written as 1519 and 7 1519. 7 (a) Find the result of this division using the standard long division algorithm. Is there a remainder in this division? (b) Based on our answer to (a), write 1519 as a product. (c) Now evaluate our answer in using long division. Write form, where a is the (d) Based on our answer to (c), write 15 as a product plus a remainder. quotient and b is the remainder of the division. Eercise #: Now let's see how this works out when we divide two polnomials. 15 18 (a) Simplif 6 polnomial long division. b performing (b) Based on our answer to (a), write 15 18 as the product of two binomial factors. emathinstruction, RED HOOK, NY 1571, 015
(c) Write 15 0 6 in the form q r 6 performing polnomial long division. Also, write a remainder., where q is a polnomial and r is a constant, b 15 18 as the product of two binomial factors plus (d) Is +6 a factor of +15 +18? Is +6 a factor +15 + 0? If a function, b() is a factor of another function, a(), the remainder when a() will be 0. b() Eercise #3: (1) Simplif each of the following epressions using polnomial long division. () Based on our answer, determine if the divisor, b() is a factor of the dividend, a(). (3) If there is a remainder, write the original problem, a() r(), in the form, q()+. This is also known as b() b() quotient-remainder form. This is the form ou used for eercise c. (a) 3-4 - + 6 + (b) 34-5 3 + + 3-3 - emathinstruction, RED HOOK, NY 1571, 015
3 (c) 43-1 + 3 + 8 +1 (d) 3-13 -1 + 3 (e) 3 +11 + 38 + 40 + 6 + 8 (f) 4 + 3 + - -1-3 emathinstruction, RED HOOK, NY 1571, 015
4 LESSON #3 - POLYNOMIAL LONG DIVISION COMMON CORE ALGEBRA II HOMEWORK FLUENCY Complete the following three steps for each problem. (1) Simplif each of the following epressions using polnomial long division. () Based on our answer, determine if the divisor, b() is a factor of the dividend, a(). (3) If there is a remainder, write the original problem, a() r(), in the form, q()+ b() b(). (a) 5 4 3 (b) 6 11 4 (c) 6 11 10 3 5 41 3 (d) 8 (e) 3 7 17 41 5 (f) 3 - - 5-1 + + 3 emathinstruction, RED HOOK, NY 1571, 015
5 (g) 3-8 - (h) 3 11 5 3. In problem 1g, ou should have had no remainder. This means that - is a factor of 3-8. How could ou have known this without doing polnomial long division? 3. Given the function f ()=3 3 +9 +8 +4 (a) Determine if (+) is a factor of f() using polnomial long division. (b) Graph f() on the grid below. (c) How does our answer to part (a) relate to the real root of the graph in part (b)? emathinstruction, RED HOOK, NY 1571, 015
6 LESSON #33 - THE REMAINDER THEOREM COMMON CORE ALGEBRA II In the previous lesson, ou learned that b() is a factor of a() if the remainder is 0 when a() is divided b b(). 3 Eercise #1: Complete the following problems given p 7 11 3. (a) Is 1 a factor of p()? (b) Write p() in factored form. (c) Graph p() on the grid below. (Do not worr about the relative minimum that we cannot see with this window. We are more interested in the roots of he graph.) (d) What is the integer root of p()? 10 (e) Evaluate p(-1). -3 10-10 Remainder Theorem Part 1: If a function is divisible b -a, then a is a root of the function. In other words, if the Remainder is zero when p() is divided b -a, then p(a)=0. It is also true that if a is a root of a function, p() is divisible b -a. emathinstruction, RED HOOK, NY 1571, 015
7 Eercise #: Complete the following problems given p( ) = 3 - -19 +10. (a) Is -5 a factor of p()? (b) Use our answer to part (a) to write p() as an equivalent product plus a Remainder. (c) Graph p() on the grid below. (c) Look at the table for p(). What is the value of p() when =5? (e) Evaluate p(5). Remainder Theorem part : The Remainder when a function, p(), is divided b -a is p(a). Look at our answer to Eercise b. The remainder theorem should make sense because p() is 15 more than where 5 would be a root of the function. emathinstruction, RED HOOK, NY 1571, 015
8 Eercise #3: Complete each of the following problems using the Remainder Theorem. (a) What is the remainder when A()=5 4-0 -84 is divided b +5? (b) What is the remainder when B()= +11 +11 is divided b +3? (c)what is the remainder when C()= 3 + -14-4 is divided b -4? Eercise #4: In Eercise #3, ou should have a Remainder of 0 in one of the three problems. (a) Use that fact and polnomial long division to completel factor the function. (b) What are the roots of the function ou chose? Verif with a graph. Eercise #5: If the ratio value of r? 11 9 was placed in the form q r, then which of the following is the 9 (1) 3 (3) 9 () 5 (4) 4 emathinstruction, RED HOOK, NY 1571, 015
9 LESSON #33 - THE REMAINDER THEOREM COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Which of the following is the remainder when the polnomial (1) 107 (3) 3 () 7 (4) 9 5 3 is divided b the binomial 8? 17 4 r. If the ratio is placed in the form q, where q is a polnomial, then which of the 5 5 following is the correct value of r? (NOTE: This is the same question as asking for the remainder of the division problem. (1) 3 (3) 18 () 177 (4) 7 3. When the polnomial p was divided b the factor 7 the result was is the value of p 7? (1) 8 (3) 11 () 7 (4) It does not eist 11. Which of the following 7 4. Which of the following binomials is a factor of the quadratic but b using the Remainder Theorem. 4 35 4? Tr to do this without factoring (1) 6 (3) 8 () 4 (4) 5. Which of the following linear epressions is a factor of the cubic polnomial 3 9 16 1? (1) 6 (3) 3 () 1 (4) emathinstruction, RED HOOK, NY 1571, 015
30 p 46 80. 6. Consider the cubic polnomial 3 (a) Using polnomial long division, write the ratio of r 3. Evaluate p 3 q p 3 in quotient-remainder form, i.e. in the form. How does this help ou check our quotient-remainder form? (b) Evaluate p 5. What does this tell ou about the binomial 5? (c) Use our answer to part (b) and polnomial long division to completel factor p(). (d) Besides 5, what are the other zeroes of p()? emathinstruction, RED HOOK, NY 1571, 015
31 LESSON #34 - USING POLYNOMIAL LONG DIVISION AND THE REMAINDER THEOREM COMMON CORE ALGEBRA II Eercise #1: For the cubic f () = 3 + 7 +13 + 3, P(-3)=0. Find the roots of the function algebraicall. Note: Two of the roots will be irrational, so ou need to choose an appropriate method to solve that part of the equation. Eercise #: When 3 +k +k +6 is divided b ( + 1), the remainder is 3. Find the value of k. Eercise #3: Given P()= 3 +k -11-6, find k if (-) is a factor of P(). emathinstruction, RED HOOK, NY 1571, 015
3 Eercise #4: The formula for the volume of a square pramid is V = 1 Bh where B is the area of the square base 3 and h is the height of the pramid. If the volume of a pramid is represented b 1 3 (3 + 9 + 4 + 0), and the height of the pramid is ( + 5), what epression represents the length of the side of the base of the pramid? This problem does not require polnomial long division. Remember, a quadratic polnomial is in the form P()= a +b +c. To find a specific quadratic function, ou need to find the values of a, b, and c. Eercise #5: Find a quadratic polnomial, P(), such that P(1) = 3, P() = 1 and P(-) = -15. emathinstruction, RED HOOK, NY 1571, 015
33 LESSON #34 - USING POLYNOMIAL LONG DIVISION AND THE REMAINDER THEOREM COMMON CORE ALGEBRA II HOMEWORK 1. For the cubic f () = 3-7 - 6, P(-)=0. Find the roots of the function algebraicall. Hint: Remember to add a quadratic term, 0, when performing polnomial long division for this problem.. If G()= 5 +k 3 - +is divided b ( - ), the remainder is 14. Find the value of k. 3. Given P()= 3 - +k +4, find k if (-1) is a factor of P(). emathinstruction, RED HOOK, NY 1571, 015
34 4. The formula for the volume of a clinder is V = pr h where r is the radius and h is the height of the clinder. If the volume of a clinder is represented b p ( 3 +8 +1 +18), and the height of the clinder is ( + ), what epression represents the radius of the clinder? 5. Find a quadratic polnomial, P(), such that P(3) = -11, P(1) = -3 and P(-) = -1. emathinstruction, RED HOOK, NY 1571, 015
35 LESSON #35 - POLYNOMIAL IDENTITIES COMMON CORE ALGEBRA II Polnomials are epressions consisting of addition and subtraction of variables and coefficients all raised to nonnegative, integer powers. As in the last few lessons, there can be a single variable or multiple variables. Eercise #1: Which of the following is not a polnomial epression? Eplain wh our choice fails to be a polnomial. (1) () 3 3 (3) (4) 7 7 1 4 Because polnomials consist of basic operations on variables, the can be manipulated using the associative, commutative, and distributive properties (as ou have done man times). These operations can result in what are known as polnomial identities. An identit is defined more broadl below: IDENTITIES An identit is an equation that is true for all values of the replacement variable or variables. WHEN PROVING AN IDENTITY, YOU MUST WORK WITH EACH SIDE OF THE EQUATION SEPERATELY. YOU CANNOT MOVE ANYTHING FROM ONE SIDE TO THE OTHER. Eercise #: One identit that ou should be familiar with is (a) Test this identit with the pair 10 and 3.. (b) Prove this identit b manipulating the right side of the equation. (c) Use this identit to evaluate the difference 50 49. (d) Use this identif to simplif and then evaluate the product 51 49. Eercise #3: Prove the identit a b a ab b and use it to evaluate 35. emathinstruction, RED HOOK, NY 1571, 015
36 Sometimes identities can have geometric connections as well as algebraic. The Pthagorean Theorem gives us ample identities. Eercise #4: A right triangle is shown below whose sides are, 1, and 1. (a) Show that these will be the side lengths of a right triangle as long as 1, i.e. show that 1 1 is an identit. (b) Based on our work from (a) and on the triangle shown, eplain wh an even integer (other than ) must be part of a Pthagorean triple, i.e. a set of 3 integers that could be the sides of a right triangle. Generate a Pthagorean triple that has 10 in it and a separate one that has 14 in it. Eercise #5: Consider the polnomial identit 3 3 3 a b a 3a b 3ab b. (a) Prove this identit b epanding the left-hand side of the equation. 3 (b) Use our calculator to find the value of 11 then use the identit to show the same result. Carefull consider our choice of a and b. emathinstruction, RED HOOK, NY 1571, 015
37 LESSON #35 - POLYNOMIAL IDENTITIES COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. One of the two epressions below is an identit and one of them is not. Determine which is an identit b testing the truth value of the equation for various values of. Show the values of that ou test. Remember, an identit will be true for ever value of. Equation #1: 1 4 4 Equation #: 4 4. Which of the following equations represents an identit? (1) 1 3 4 (3) 4 3 7 (4) () 6 3 10 4 5 0 8 3. One of the more useful identities that students almost inherentl learn is: c d c d cd (a) Prove this identit. You ma choose to algebraicall manipulate one or both sides of the equation to justif the equivalence. (b) This identit allows ou to multipl common binomials ver quickl. Find the following products in simplest trinomial form. (i) 3 7 (ii) 7 (iii) 10 3 emathinstruction, RED HOOK, NY 1571, 015
4. You should be well aware of the difference of perfect squares, i.e. an identit for the difference of perfect cubes: 3 3 38. But there is also (a) Prove this identit b epanding the product on the right-hand side of the equation. 3 3 (b) Use the identit to find the value of 10 9 without the use of our calculator. Show the steps in our calculation. Then, verif with our calculator. APPLICATIONS 5. Another famous identit that can be used to generate Pthagorean Triples is shown below: The complicated sides are shown on the diagram. (a) Prove this identit b epanding the products on both sides of the equation. (b) Generate the sides of the right triangle if 4 and 1. Show that these sides satisf the Pthagorean Theorem. (c) Reasoning: This relationship leads to the conclusion that there is no Pthagorean triple of this form that contains the integer. Wh? emathinstruction, RED HOOK, NY 1571, 015
39 LESSON #36 POLYNOMIAL WORD PROBLEMS DAY 1 COMMON CORE ALGEBRA II Eercise #1: f ( ) ( )( 1)( 5). 50 a) What is the -intercept of f()? b) What are the zeroes of f()? -3 6 c) Is f() increasing or decreasing over the interval -3<<-1? -50 d) Is f() positive or negative over the interval 1<<5? e) How man relative etrema does f() have? Eercise #1: Using the diagram to the right, f ( ) ( 1)( 4) models the volume of the bo where is measured in inches. Let = Let f() = The volume of the open bo is 40 cubic inches. Determine algebraicall the dimensions of the bo. Verif our answer using a graph. f() emathinstruction, RED HOOK, NY 1571, 015
Eercise #3: A rectangular bo has a height of feet. Its width is three more than its height, and its length is one less than twice its height. If the volume of the bo is 90 cubic feet, use a graph to find the dimensions of the bo. Write a function, f() to model the volume of the bo first. 40 10 f() 5 Eercise #4: The weight of an ideal round-cut diamond can be modeled b where w is the diamond s weight in carats and d is its diameter in millimeters. 3 w 0.0071 d 0.090d 0.48d a) According to the model, what is the weight of a diamond with a diameter of 15 millimeters to the nearest thousandth of a carat? - 0 b) What is the ideal diameter of a carat round-cut diamond that is carats, what diameter would the ideal round-cut diamond to the nearest hundredth of a millimeter? emathinstruction, RED HOOK, NY 1571, 015
41 LESSON #36 - VOLUME WORD PROBLEMS COMMON CORE ALGEBRA II HOMEWORK 1. f ( ) ( 5)( 3)( 6). a) What is the -intercept of f()? 00 f() b) What are the zeroes of f()? c) Is f() increasing or decreasing over the interval -<<1? -7 7 d) Is f() positive or negative over the interval -<<1? -50 e) How man relative etrema does f() have?. Using the diagram to the right, the volume of the open bo can be modeled b the function, f ( ) 3 ( 1)( 3) where is measured in inches. Let = -1 Let f() = 3 The volume of the open bo shown to the right is 36 cubic inches. Determine algebraicall the dimensions of the bo. Verif our answer using a graph. -3 f() emathinstruction, RED HOOK, NY 1571, 015
3. You are designing a rectangular swimming pool that is to be set into the ground. The width of the pool is 5 feet more than the depth and the length is 35 feet more than the depth. a) Write a function, V() to model the volume of the pool where represents the depth of the pool in feet. 4 Let = Let V() = 3000-5 10 b) If the depth of the pool is 6 feet, how much water does it hold? -000 c) If the pool holds 000 cubic feet of water, what are the dimensions of the pool? Determine our answer with a graph. emathinstruction, RED HOOK, NY 1571, 015
43 LESSON #37 - OTHER POLYNOMIAL WORD PROBLEMS COMMON CORE ALGEBRA II Eercise #1: The profit P (in millions of dollars) for a T-shirt manufacturer can be modeled b P = - 3 +4 + where is the number of t-shirts produced (in millions). The manufacturer has the abilit to make up to 5 million t-shirts. Let = Let P = a) How man t-shirts should the compan make to maimize their profit. Round to the nearest thousand T-shirts. b) What is the maimum profit to the nearest thousand dollars? c) For what numbers of t-shirts produced is the profit decreasing? Round the boundaries of the interval to the nearest ten thousand t-shirts. d) What is the average rate of change in the profit when the compan goes from making 1.5 million t-shirts to 3 million t-shirts? e) For what numbers of t-shirts produced does the compan make mone? Epress our answer to the nearest hundred t-shirts. f) Currentl, the compan produces 4 million t-shirts and makes a profit of $4,000,000. Determine algebraicall a lesser number of t-shirts the compan could produce and still make the same profit. Check our answer on the graph. emathinstruction, RED HOOK, NY 1571, 015
44 Eercise #: Jessica started a new bracelet compan. Her net monthl profit in hundreds of dollars can be 3 modeled b the function h( ) 18 9 10 where is the number of months since she started the compan. a) Graph Jessica s profit function on the aes provided using the domain 0 1. Choose an appropriate scale. b) h(5) = 15. Eplain what this means in the contet of the problem. c) In what month did Jessica make the most mone? d) During what month(s) did her profit decrease? Round to the nearest tenth of a month. e) Do ou think g() will model Jessica s profits over the net ear, when is an integer such that 13 4? Eplain our answer. emathinstruction, RED HOOK, NY 1571, 015
45 LESSON #37 - OTHER POLYNOMIAL WORD PROBLEMS COMMON CORE ALGEBRA II HOMEWORK 1. The profit, P (in millions of dollars) for a manufacturer of cellphones can be modeled b 3 P 4 1 16 where is the number of cellphones produced (in millions). The compan has the abilit to make 5 million cell phones. Let = Let P = a) How man cellphones should the compan make to maimize their profit. Round to the nearest thousand cellphones. b) What is the maimum profit to the nearest thousand dollars. c) For what numbers of cell phones produced is the profit increasing? Round to the nearest hundred cell phones. d) What is the average rate of change in the profit when the compan goes from making 500,000 cell phones to,500,000 cell phones? e) For what numbers of cell phones produced does the compan lose mone? f) Currentl, the compan produces 3 million cellphones and makes a profit of $48,000,000. Determine algebraicall a lesser number of cell phones the compan could produce and still make the same profit. Check our answer on the graph. emathinstruction, RED HOOK, NY 1571, 015
46. For the 1 ears that a grocer store has been open, its annual revenue R (in millions of dollars) can be 4 3 modeled b the function R( t) 0.0001( t 1t 77t 600t 13,650) where t is the number of ears since the store opened. a) Graph R(t) on the aes provided Choose an appropriate scale. R(t) t b) R(8)=1.557. Eplain what this means in the contet of the problem. c) In which ear(s) was the revenue $1.5 million? d) During what ear(s) was the compan s profit increasing? Round to the nearest tenth of a ear. emathinstruction, RED HOOK, NY 1571, 015
47 LESSON #38 ADVANCED FACTORING COMMON CORE ALGEBRA II This lesson describes two methods of factoring which build on methods alread practiced in this unit. Factoring using U-substitution: You will be given a polnomial in the form of a factorable quadratic: au² + bu + c, where U can be an function of. You ma be asked to factor such an epression, or to solve an equation (for ). Follow these steps: 1) Determine the function of, and set U = that function ) Subsitite U into the equation for the function of. 3) Factor using U as the variable 4) Substitute the given function of back in for U. 5) Factor again if possible and/or solve for if it is an equation. Eercise 1: Factor completel: 4 5 5 4 5 6 Eercise : Solve 9 9 35 0 algebraicall. Verif our results graphicall. emathinstruction, RED HOOK, NY 1571, 015
Factoring polnomials containing 6 terms grouping method Sometimes the method of factoring b grouping works with polnomials containing 6 terms. To factor polnomials of this tpe, ou can tr to split the polnomial into 3 groups of, or if that doesn t work, split it into groups of 3. As alwas, continue factoring if possible after factoring b grouping. Eercise 3: Factor completel: ⁵ + ⁴ ³ ² 4 48 Eercise 4: Factor completel: ³ + 3² 5 + 8² + 1 0 Eercise #5: Advanced Factoring Practice a) Factor completel: a² + 3a + a + b² + 3b + b emathinstruction, RED HOOK, NY 1571, 015
49 b) Solve for all values of : ( +0) +( +0)-0= 0 c) Factor completel: (3 +) -13(3 +)+40 d) Factor Completel: - -5 +10-6 +1 emathinstruction, RED HOOK, NY 1571, 015
50 1. Factor completel: 8 1 LESSON #38 - ADVANCED FACTORING COMMON CORE ALGEBRA II HOMEWORK. Solve for all values of : 11 4 0 3. Solve algebraicall for all values of : ⁵ + ⁴ + ³ + ² 1 1 = 0 4. Factor completel: 6a 15a + b 5b 4c + 10c emathinstruction, RED HOOK, NY 1571, 015
51 5. Solve for all values of : 7 8 6. Factor completel: 4 3 7. Factor completel: 8ab ac + 4ad + 1b 3c + 6d 8. Solve algebraicall for all values of : 3⁴ + ³ ² 1² 8 + 4 = 0 emathinstruction, RED HOOK, NY 1571, 015