POLYNOMIALS and FACTORING


 Ami Cunningham
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1 POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use when simplifying expressions involving exponents? 3. Use the power rule for exponents, 4. Use the power rules for products and quotients, 5. Use the quotient rule for exponents, and define a number raised to the 0 power 6. Be able to decide which rule to use Evaluate exponential expressions Begin with simple exponents: eg.. 3,3,,, and ask students to evaluate What happens when the negative is there? eg.. 3,, What happens when you have ( 3 ) 3, ( ) 3, ( ) 4? What is the difference between the two examples? Use the product rule for exponents, m n m n a a = a + Then look at what happens when you multiply the same bases: eg.. 3 3, Have students work through several examples asking whether they see a pattern. Does a pattern exist? Have students work through that = = How about eg.. a a, b b? What pattern do you notice and how can you represent that pattern? m n m n Have students draw to the conclusion that a a = a +
2 3 4 m Use the power rule for exponents, ( ) n mn 3 a = a Now try examples involving powers: eg.. ( 3 ),( ) What does the 4 mean in ( 3 ) 4 3 How is this similar to 3 3 example from before? Have students work through several examples asking whether they see a pattern. Does a pattern exist? How about ( ),( ) a b? What pattern do you notice and how can you represent that pattern? m Have students draw to the conclusion that ( ) n Use the power rules for products ( ab) n n n = a b Now try examples involving powers: eg.. ( 3 ),( 4 ) 3 4 What if you have ( ab),( xy), or ( 3x) instead? a = a mn 3 Students do several examples asking whether they see a pattern. Can we use our other rules to help us? Does a pattern exist? What pattern do you notice and how can you represent that pattern? Have students draw to the conclusion that ( ) n n n ab = a b a a Use the power rules for quotients, = b b n n n Now try examples involving powers: eg.. 3 Do several examples asking students whether they see a pattern. Does a pattern exist? a x Ask them whether it would be try for,? b y What pattern do you notice and can you represent that pattern? a a Have students draw to the conclusion that = b b 3 3 n 4 n n
3 Use the quotient rule for exponents, define a number raised to the 0 power Be able to decide which rule to use a a m n a m n =, and Begin with the quotient rule, but use the same base. Do several examples asking students whether they see a pattern. Does a pattern exist? What pattern do you notice and can you represent that pattern? m a m n Have students draw to the conclusion that = a n a Can you define what it means to be raised to the 0 power? Help students define Begin with an example like: ( x y ),3 ( z ) Is there a strategy for picking which rule to use? Explain how you would decide which rule to use? Give worksheet on more difficult problems. Students can work in groups Ask students to talk through their strategy on how to solve. Ask students if they performed them in a different way? Is it correct to say there are many ways to simplify? Why and explain? Negative Exponents and Scientific Notation ( days); 1. Simplify expressions containing negative exponents. Use the rules and definitions for exponents to simplify exponential expressions 1. What does it mean to have a negative exponent?. How can you represent very small or very large numbers using exponents? 3. Write numbers in scientific notation 4. Convert numbers in scientific notation to standard form
4 Simplify expressions containing negative exponents Use the rules and definitions for exponents to simplify exponential expressions m a m n = a n Look at the rule: a What happens to the expression when n is greater than m? What does the fraction look like? Where does the base end up? What happens when the negative exponent is in the denominator? Handout of worksheet for students to simplify expressions that also involve negative exponents. When complete ask students to explain how they simplified. Ask if anyone did it differently and ask them to explain their method. Write numbers in scientific notation Can you represent very large numbers using exponents?,100,000,000,000,000,000. Start with 310. Is this equal to 3.1 x 10? Then use 3.1 x 10 1 Can you represent 3100 by 3.1 x 100? Then use 3.1 x 10 How would you present using this pattern? Then try 3,100,000,000. What number would x 10 6 be? Now look at Is this equal to 4.4 x 101? Can you represent using this type of pattern? Is there a general strategy for writing small or large numbers using exponents? Convert numbers in scientific notation to standard form Can we extend strategies learned with exponents and scientific notation to simplify these expressions? ( x 10 6 )(3 x 103 ) 10 8x10 How about expressions like these? How would you simplify these expressions? 3 4x10 Introduction to Polynomials (1 day) 1. Define term and coefficient of a term. Define polynomial, monomial, binomial, trinomial, and 1. What is the difference between a monomial, binomial, trinomial, etc.?. What is a like term?
5 degree 3. Evaluate polynomials for given replacement values 4. Simplify a polynomial by combining like terms 3. Why would you want to simplify a polynomial? 4. When do you simplify a polynomial? 5. How do you know whether a polynomial is considered simplified? 5. Simplify a polynomial in several variables Define term and coefficient of a term Provide students the definitions for term and coefficient. Some examples should contain one variable only and others with several variables. Give students several examples to talk through. What is a constant term? What does the word constant mean? What is a numerical coefficient? What can these terms represent? Define polynomial, monomial, binomial, trinomial, and degree Ask students to list words having the prefixes mono, bi, tri, and poly. Allow students to discuss the meanings of the prefixes based on the words on their lists. Provide several examples containing a monomial, binomial, trinomial, and higher degree polynomials. Ask students to apply the meanings of the prefixes to guess the definitions of monomial, binomial, trinomial, and polynomial. What difference do you notice with these examples? What is a polynomial? If we call the two term polynomial a Binomial, what do you think we call a three term polynomial? How do you know what is the degree of the polynomial? Evaluate polynomials for given replacement values How do you evaluate a polynomial given replacement values? Have students work in groups to evaluate polynomials. Give several word problems like Finding Freefall Time from a building. A building is feet tall. An object is dropped from the highest point. Neglecting air resistance, the height in feet of the object above ground at time t seconds is given by 16t Find the height of the object when t = 1 second, and when t = 3 seconds, etc. What do these answers mean? Ask students to think about how the given equation relates to the problem.
6 What does represent in the equation? What if the building was 63.7 feet tall? How would the equation differ? Do you think this polynomial will give a good estimate of the height of the object for all values of t? They should say no, because once the object hits the ground the polynomial does not apply. Simplify a polynomial by combining like terms What is a like term? Give several examples: {5x, 7x }, {a b 3, 3ab 3 }, etc. Which of those are like terms? How do you think you can simplify terms that are alike? What if you have 3 squares + 4 squares? Do you have 7 squares? Can you combine 7 squares + 3 watermelons? What are the like terms? Simplify a polynomial in several variables Recall that like terms may have several variables. Have students work in groups to combine several polynomials involving single variables and multiple variables. What did everyone get for each answer and have student discuss their findings. Adding and Subtracting Polynomials (1 day) 1. Add polynomials. Subtract polynomials 1. How do you add and/or subtract polynomials?. How do you know you are finished? 3. Add or subtract polynomials in one variable 4. Add or subtract polynomials in several variables Add polynomials Recall what it means to be like terms. Look at several examples of addition. For example (3x 5 7x 3 + x 1) + (3x 3 x) Ask students to combine like terms.
7 Have them discuss their answers. If they have differences, have them work out which is correct and why. Subtract polynomials What do you think happens when you subtract? (3x 5 7x 3 + x 1)  (3x 3 x) Think about 75 (6+). Are you subtracting 8 or just 6? So, you subtract the entire parenthesis? How about 75 (8 3)? Are you subtracting out 8? 3? 5? Which one? What do we do if we introduce variables? Have students draw to the conclusion that the minus sign must be distributed. What do we mean when we are subtracting polynomials? Add or subtract polynomials in one variable Pose this question to the group and have the groups create the problem and simplify: Subtract the sum of (3x+6) and (8x5) from (5x+) Have students provide their answer. If there are discrepancies, have students determine which is correct. Is there a method to which you simplify these expressions? Add or subtract polynomials in several variables Based upon what we ve learned, is there going to be any difference in how you simplify problems with several variables? If any, what is that difference? (a ab + 6b ) + (3a + ab 7b ) (5x y + 3 9x y + y ) ( x y + 7 8xy + y ) Multiplying Polynomials (1 day) Multiply monomials Multiply a monomial by a polynomial 1. How do you multiply polynomials?. Why do you FOIL when you have two binomials? Multiply two polynomials Multiply polynomials vertically Multiply monomials Recall how to multiply monomials with the same base: x * x 3 = x 4 or 6a *7a 3 b = 4a 5 b
8 Multiply a monomial by a polynomial How would you multiply 7(6+3+)? What is the answer? Did you multiply 7 to the entire summation or to 6 only? Have students come to a consensus that you multiply 7 to the entire summation. How would you apply this same principle to 7(x+), 7x(x+), 7x(3x + x + )? Have students come to the conclusion that you multiply the monomial to each term inside the parenthesis. Always remember to simplify as much as possible when you ve multiplied. For example: 3x (x+3) + x (x  5) = 6x 3 +9x +x 5x Are we done yet? No. Can you simplify? Yes, it equals 6x 3 +11x 5x Multiply two polynomials Recall when multiplying, we had to multiply the monomial to EVERY term inside the parenthesis. What do you think happens when we have a binomial multiplied to another binomial? (x+3)(x) Have students talk about what they think you should do to multiply this out. Have them come to the conclusion that you should multiply each term inside one parenthesis to each term in the next parenthesis. How would it change if we had two variables in there? (x+3y)(xy) Have students come to the conclusion that it does not change the way they multiply. Multiply polynomials vertically and/or the box method. Show students the box method: (3x+)(x5) 3x x 6x 4x 515x 10 Show students the vertical method. Show them the same example so they see the result is the same. Give a worksheet with several examples to work with in groups. Have them try all three methods: box, vertical, and distributive methods. In whole class discussion, have students talk about which method they prefer and why. Special Products (3 days) 1. Multiply two binomials using the FOIL method 1. Why don t you just square the two terms when you have a square of a binomial?
9 . Square of a binomial. How do you recognize these special products? 3. Multiply the sum and difference of two terms 4. Use special products to multiply binomials Multiply two binomials using the FOIL method We ve seen the box method, vertical method and the distributive property method, let s look at the FOIL method. Once you become familiar with all four, you can use which ever method you prefer. Point out that special products are shortcuts for multiplying binomials they can STILL be worked out using the methods seen earlier. Square of a binomial Have student multiply (4x+3). Then have them multiply (4x3). What is the relationship between the problem and its solution? Do you recognize a pattern? Return to the box method: (a+b)(a+b) a b a a ab b ab b Can you recognize a pattern? How about this: (x+3) x 3 x x 3x 3 3x 3 How about this: (3x+y)(3x+y) 3x y 3x (3x) 3xy y 3xy y Can you recognize a pattern? Have students come to the conclusion that (a+b) = a +ab +b Multiply the sum and difference of two terms Have student multiply (4x3) (4x+3). What about (x+y)(xy)? What is the relationship between the problem and its solution? Do you recognize a pattern? Return to the box method: (a+b)(ab) = a b a b a a ab
10 b ab b Can you recognize a pattern? How about this: (x+3)(x3) = x  9 x 3 x x 3x 33x 3 How about this: (3x+y)(3xy) = 9x y 3x y 3x (3x) 3xy y 3xy y Can you recognize a pattern? Have students come to the conclusion that (a+b)(ab) = a  b Use special products to multiply binomials Give students worksheets to work in groups. Have students report their findings/answers. If there are any differences in answers, have student discuss which is correct. Dividing Polynomials ( days) 1. Divide a polynomial by a monomial, a + b a b = +, where c 0 c c c 1. Why do you need a common factor in each term to cross out?. What are the similarities between using long division in arithmetic versus variables?. Use long division by a polynomial other than a monomial Divide a polynomial by a monomial x Can you divide? Why or why not? What about? What is the difference 0 x + 3 between the two examples? Why can you divide one of them to get 0, but not the other?
11 How do you remember which is which? Have students explain their rationale. Let s look at an example where the denominator is not equal to 0. How can you simplify 3x + 6? Have students guess. If they guess to just cross out the 3, to equal x+6, ask them is = = + 6 = 8? Why or why not? Have students come to the conclusion 3 3 that you must cross out factor of 3 from EVERY term. 3 x + 6 = x + 3 Check their answer by plugging in x = 1,, and 3, into the original expression as well as the resultant to make sure they match. Point out that simplifying is not CHANGING the result it just makes it look prettier and SIMPLIER Have students work through several examples in pairs and discuss the results. Have students come to a general consensus on the correct answers. The Greatest Common Factor (1 day) Find the GCF of a list of numbers Find the GCF of a list of terms Factor out the GCF from the terms of a polynomial 1. Why do you need to find the GCF?. How can you check to see if you factored correctly? 3. Is it factored completely? Factor by grouping Find the GCF of a list of numbers Point out to students that they already have the tools to solve linear equations such as 8x = x 1, but have not learned the tools to involve equations in involving higher order polynomials such as x +8x = x1. The factoring skills students learn in the remaining sections will give them the tools needed to begin solving these more complicated kinds of equations. Recall multiplying out 5(x+) = 5x+10. This is multiplying.
12 NOW, we will look at 5x + 10 = 5(x+). This is factoring. (Draw a diagram to illustrate the reverse nature) BUT FIRST, we have to look at something called the Greatest Common Factor. In the product 3*5 = 15, 3 and 5 are called factors and 3*5 is called the factored form of 15. What are the factors of 4, 14, 8? What do these numbers have in common? What do you think GREATEST Common Factor means? What would be the GCF of 4, 14, 8? What would be the GCF of 4 and 8? How about 60 and 4? Find the GCF of a list of terms Let s extend this to variables. What does x 3 mean? Have students conclude that it is x*x*x. x is a factor and there are 3 of them. Ask students to talk about how they would find the GCF for y 4 and y 6. What is the GCF of 1 and 3x? What is the GCF of 6x, 9x 4, and 1x 5? Have students work through a worksheet with 5 different examples and discuss their results. Have all the students come to a consensus on the correct answers. Factor out the GCF from the terms of a polynomial From the example above, how can you factor 6x +9x 41x 5 knowing the GCF? Factor 10x 3 + 8x  x. Point out that we don t like a () sign in the front of a trinomial. How would you factor this polynomial? Give students worksheet for group work. Have all the students come to a consensus on the correct answers. Ask students to construct a trinomial whose GCF is 4y 3. Have students discuss their trinomials and ask them if 4y 3 is indeed the GCF. Point out that there are many answers to this. Factor by grouping Give students an example like xy + 5y 4x 10y. Ask if they can see a GCF for this example. Have them talk about the ways you could group terms together to form GCF for those groups. When they are about to find the following: y(x + 5y) (x + 5y), ask what do you notice? Can you factor that some more? Why or why not? What would it become? (y)(x+5y) Try another example 16x 3 8x  1x 1. Answer: (4x7)(4x +3) This is called Factoring by Grouping  since you grouped terms together that have GCF Explore those answers where students group incorrectly!! You must have a common factor to be able to factor completely. Give a worksheet for students to work on is groups. Discuss results for a consensus on the answers.
13 Factoring Trinomials of the Form x + bx + c (1 day) 1. Factor trinomials of the form x + bx + c by unfoiling. Factor out the GCF and then factor trinomials of the form x + bx + c 1. What does it mean to factor?. What is the end result when you factor? 3. What is the difference between multiplying polynomials and factoring polynomials? 4. How can you check to see if you factored correctly? 5. Does every polynomial factor? Why or why not? 6. Has it been factored completely? Factor trinomials of the form x + bx + c Factor out the GCF and then factor trinomials of the form by unfoiling Encourage students to look at this section as brainteasers to solve. If we multiplied two binomials and received a trinomial, is there a way to get the two binomials by factoring a trinomial? Look at x x 1. Think about how to factor. Have students discuss in groups. Elicit answers and ask how did you get that? Did you check to see if you factoring correctly? How could you check? Ask if someone else got it a different way. Emphasize the correct methods. Do the same for another example like x 7x Show the backwards box: x a x (x) ax b bx ab Recall when multiplying binomials, ax + bx = the middle term. Ab = constant term So, a + b = 7 and ab = 50. (x 5)(x) What are other ways you can factor this? Can you recognize a pattern? Have students work through another example: y 3y 40 and q +6q + 9. Talk about the signs of these numbers and the significance in the answer. Try another example: x + 6x + 15 this is prime! Whenever you ask students to factor, always ask is it factored completely? Try another problem: 4x  4x + 36 What is the difference with this problem? Make sure
14 students notice the 4 in front. What could you do with the 4? Once you factor the 4, 4(x  6x + 9), now what do you do? Is it factored completely? Have students continue to factor so 4(x 3)(x 3) or 4(x 3) Try another problem: x 3 + 3x 4x. Have students discuss how they would factor. After factoring, ask is it factored completed? How do you know it s factored completely? Ask how they factored it? Elicit other ways from students. Discuss a pattern of how they factored. First, factored out GCF, then they factored the trinomial. Factoring Trinomials of the Form ax + bx + c Factor trinomials of the form ax + bx + c, where a 1 ( days) 1. What changes in how you factor when a 1. How do you remember to always check for a GCF before factoring any trinomial? Factor out the GCF before factoring a trinomial of the form ax + bx + c Factor trinomials of the form ax + bx + c, where a 1 3. How can you check your final answer? 4. How do you know your polynomial is completely factored? With a problem like 4x  4x + 36, the 4 was a GCF. What if you don t have a GCF? How can you factor it? For example: 3r +10r 8, answer: (3r )(r + 4) Ask students to think about how they would factor it? Discuss different factoring ideas. Ask students to get into groups to talk about ways of factoring. How did they factor their trinomial? Have students discuss the logic behind their method of factoring. Is it factored completely? Why or why not? Is (3r)(r+4) = 3r (r+4)? NO, Don t make that mistake!!! Always put your parenthesis correctly. Try another one: 35x +4x 4. Answer: (5x + )(7x ) Try another one: 6x 4 5x + 1. Answer: (3x 1)(x 1). What was the difference with this one? Is it factored completely? Try one more: 1a 16ab 3b. Answer: (6a + b)(a 3b)
15 Factor out the GCF before factoring a trinomial of the form ax + bx + c What are the types of strategies we used so far? o Factoring out a GCF if there is a factor in common of all the terms. o If a = 1, using the reverse box method (unfoil) o If a not equal to 1, (list methods students were using) What would you do if you had a trinomial like this: 3x x + 10x What is the difference with that trinomial? (each term has an x) Elicit methods students would use to factor. Ask students to factor and report their answers. Talk through how they got their answer. Did anyone else factor differently? Did you get the same answer? Why or why not? Try another problem: 6xy + 33xy 18x. This will take more time to answer 3x(y1)(y+6) Try another problem: 5x 19x + 4. Answer: (x+4)(5x1) Ask students complete worksheets in groups. Report answers and have students come to a consensus on answers. Factoring Trinomials of the Form ax + bx + c 1. Use the grouping method to factor trinomials of the form ax + bx + c by Grouping ( days) 1. Why would you want to factor by grouping?. When would you want to factor by grouping? Use the grouping method to factor trinomials of the form ax + bx + c Recall before when we had four terms and we grouped terms together in order to factor? Recall 16x 3 8x  1x 1. Answer: (4x7)(4x +3) Can we also use this method in factoring trinomials? How do you think we could use this? How could you separate the b so you can factor by grouping? Try this example: 3x +14x + 8 Show them this method: Factors of (3 * 8) = 4 Sums of the factor (sum to 14) 1, 4 5,1 14
16 3, x +14x + 8 = 3x + 1x + x + 8. How do you know which terms should be grouped together? Then, factor by grouping. 3x(x+4) + (x+4). Is it done? Answer: (3x + )(x + 4) Can you think of another way to factor trinomials by grouping? What are they? Factor: 30x  6x + 4. Factor: 6x y 7xy 5y (Here they have to remember to factor out that y or else it is not factored completely. Is it factored completely? Give students worksheets and discuss answers and methods students used to completely factor the trinomials. Factoring Perfect Square Trinomials and the Difference of Two Squares (1 day) Recognize perfect square trinomials Factor perfect square trinomials Factor the difference of two squares 1. How can you recognize a perfect square in order to factor?. How can you recognize a difference of two squares? Recognize perfect square trinomials Factor perfect square trinomials Ask students to factor x + 1x +36, x + 0x + 100, and x + 18x + 81 and report their answers. x + 1x +36 = (x+6), x + 0x = (x+10), and x + 18x + 81 = (x+9) What is the pattern? What is the relationship between the constant terms on each side of the equations? 36 = 6 ; 100 = 10 ; 81 = 9 ; What is the relationship between the middle term and the constant term on the other side? 1 = (6); 0 = (10); 18 = (9) Recall when we multiplied binomials like: (a+b) = a +ab +b or (x + 3) = x +6x+9 It would be greatly beneficial for students to be able to recognize Perfect Squares or Difference of two Squares. Ask students to name the Perfect squares: 1, 4, 9, 16, 5, 36, 49,
17 64, 81, 100, Remembering these will be helpful. Try factoring x 10x + 5. Answer: (x 5) Why do we SUBTRACT 5 here? Try factoring 9r +4rs + 16s. Can you use difference of squares here? Why or why not? Answer: (3r + 4s) Factor 1x 3 84x + 147x; answer: 3x(x7) Factor the difference of two squares Factor 4x + 9. What did you get? It s prime. If students did give you an answer other than itself, have them explain their answer and reasoning. Have students comment on why they are incorrect constructively. How would we factor 4x 9? Recall when we multiplied: (3x+y)(3xy) We recognized the pattern as (a+b)(ab) = a  b How could you use this information to factor 4x 9? (x3)(x+3) Try factoring 9x 5 5x 3 ; What do you notice here that could help you factor? Answer: x 3 (3x 5)(3x+5) Try factoring 9x Have students talk about how they would factor. If recognize that they have to factor out the negative first, Answer: (3x10)(3x+10) If they don t remember that, have them reorder the terms 100 9x ; answer (10 3x)(10+3x) Is it factored completely? Give students a worksheet to work in groups. This worksheet should include all the possible factoring problems. Discuss answers. Have students give answers and defend their answers. Always ask is it factored completely? Solving Quadratic Equations by Factoring ( days) 1. Solve quadratic equations by factoring. Solve equations with degree greater than by factoring 1. What does it mean to solve ax bx c. How many answers should you get? + + = 0 3. How do you recognize that solving this equation, xintercepts, and the zeros are the same? 4. Why do you factor and set each factor equal to zero? 5. Does you answer make sense?
18 6. Here we expect SOLUTIONS since there is an equal sign. Solve quadratic equations by factoring Ask students to suppose they are told that the product of two factors is 1. Do they know for certain what either factor is? Why or why not? Now, the product of two factors is 0. Do you know for certain what either factor is? Ask if you have an equation like ab = 0, what can a and b equal? Have students talk about the implications on whether a = 0 and/or b = 0. How could you solve an equation like (x + ) (x3) = 0? {, 3} Why would you set each factor equal to zero? What would be the solution to (x10)(3x+1) = 0? {10, 1/3} What would you do if you have an equal like x 4x+3 = 0? How could you solve this equation? {1,3} How could you check your answer? What is the solution to x(3x+7)=6? How would you solve this equation? What is the first thing you should do? Why? {/3, 3} How many answers should you get? How many answers do you expect to get when you have a linear equation? Does the degree of the equation have anything to do with the expected number of solutions? Why or why not? What do you think the connection is? Solve equations with degree greater than by factoring Try solving x 3 18x = 0. How many solutions should you get? (3) {0, 3, 3} Whenever you have an equations with a degree higher than 1, you should factor. Always factor. Solve (x + 3)(3x 0x 7) = 0; { 3, 1/3, 7} Have students talk about how they solve this. How would you solve this equation? Why? Quadratic Equations and Problem Solving ( days) 1. Solve problems that can be modeled by quadratic equations. 1. When will the ball hit the ground?. What type of problems can be represented using a quadratic equation? 3. Does your answer make sense? Can you justify it as being correct?
19 Solve problems that can be modeled by quadratic equations. Look a ball projectile problem. Since students haven t graphed yet, graph it to show them the projectile of the ball. Have students discuss the meaning behind the graph. When would the height of the ball be at its peak? When would the ball hit the ground? Where the ball hits the ground, what do we call that on the graph?
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