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Polynomial Operations and Classification

This lesson covers the classification of polynomials, standard form, degree, and basic operations including addition, subtraction, multiplication, and division. It also introduces the concept of closure and demonstrates polynomial division using both long division and synthetic division methods.

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Polynomial Operations and Classification

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  1. Warm up 4/13/2017 • Polynomials can be classified by the number terms as well as by the degree of the polynomial. Complete the following chart (we will go over this if you don’t remember.)

  2. Unit 6: Polynomial operations Test 4/26

  3. CONCEPTS/SKILLS TO MAINTAIN • It is expected that you have prior knowledge/experience related to these concepts and skills identified below. It may be necessary to attend tutoring if you have trouble with these concepts in order to build on them in this unit: • Combining like terms and simplifying expressions • Long division • The distributive property • The zero product property • Properties of exponents • Simplifying radicals with positive and negative radicands • Factoring quadratic expressions

  4. What is a POLYNOMIAL Function? • A polynomial is the monomial or the sum of monomialswith all exponents as whole numbers and coefficients are all real numbers. • Ex x3+6x2+12x+8

  5. Is it a polynomial? YES a) NO b) NO c) YES d)

  6. vocabulary • Coefficient: a number multiplied by a variable. • Pascal’s Triangle: an arrangement of the values of nCrin a triangular pattern where each row corresponds to a value of n • Polynomial: amathematical expression involving a sum of nonnegative integer powers in one or more variables multiplied by coefficients.

  7. STANDARD FORM:The terms of a polynomial are in STANDARD FORM when they are ordered from left to right in decreasing order; which means from the largest exponent to the smallest. DEGREE:The largest exponent in the polynomial. It determines the number of zeroes. Classifying: x0 x1 x2 x3 x4 0 – Constant 2 – Quadratic 4 - Quartic 1 – Linear 3 – Cubic LEADING COEFFICIENT:The first coefficient once in standard form.

  8. Basic arithmetic with polynomials • Bob owns a small music store. He keeps inventory on his xylophones by using x2 to represent his professional grade xylophones, x to represent xylophones he sells for recreational use, and constants to represent the number of xylophone instruction manuals he keeps in stock. • A) If the polynomial 5x2 + 2x + 4 represents what he has on display in his shop and the polynomial 3x2 + 6x + 1 represents what he has stocked in the back of his shop, what is the polynomial expression that represents the entire inventory he currently has in stock? • B) Suppose a band director makes an order for 6 professional grade xylophones, 13 recreational xylophones and 5 instruction manuals. What polynomial expression would represent Bob’s inventory after he processes this order? Explain the meaning of each term.

  9. Find the following products. Be sure to simplify results.

  10. ESSENTIAL QUESTIONS • How can we write a polynomial in standard form? • How can we write a polynomial in factored form? • How do we add, subtract, multiply, and divide polynomials? • In which operations does closure apply? • How can we apply Pascal’s Triangle to expand (x + y)n • How can you find the inverse of a simple function?

  11. EVIDENCE OF LEARNING • By the conclusion of this unit, you will be able to demonstrate the following competencies: • perform operations on polynomials (addition, subtraction multiplication, long division, and synthetic division) • identify and which operations are closed under polynomials and explain why • write polynomials in standard and factored forms • perform binomial expansion by applying Pascal’s Triangle • find the inverse of simple functions and verify inverses with the original function

  12. closure property • Remember: A system is closed under an operation if all the results of the operation applied to the set are contained within the set. • For example: • The set of integers is closed under the operation of addition because the sum of any two integers is always another integer and is therefore in the set of integers. • Is the set of integers is closedunder the operation of division? • Do you think that polynomial addition and subtraction is closed? • Is polynomial multiplication is closed?

  13. Polynomial Division • There are 2 ways of dividing polynomials – one of them is MUCH easier BUT doesn’t always work… sorry! • method 1 – LONG DIVISION • method 2 – SYNTHETIC DIVISION

  14. Polynomial long Division

  15. Polynomial long Division

  16. Polynomial Division • As you can see, long division can be quite tedious. However, when the divisor is linear, there is a short cut. • Let us consider another way to show this division called synthetic division. • The labor of dividing a polynomial by x – t can be reduced considerably by eliminating the symbols that occur repetitiously in the procedure. Let us consider the following division.

  17. Synthetic division

  18. Synthetic division

  19. Synthetic division • There are a few things to consider. The number in the upper left-hand corner is t, if we are dividing by x – t. What would be the divisor if we are dividing by x+ t ? • The top row consists of the coefficients of the terms of the dividend polynomial in descending order. Since the order of the coefficients denotes its corresponding power of x what do you think happens if the dividend is missing a term in the sequence? For instance, how would we represent as a dividend in synthetic division?

  20. Warm up 4/14/2017 • Factor the following polynomials: • 6x2-13x-5 • 6x2-7x-3 • 14x2-9x+1 • x2-33x+32 • x2+15x+50

  21. Is it a polynomial??

  22. Divide these polynomials using synthetic division • (11x + 20x2 + 12x3 + 2) ÷ (3x + 2)

  23. Divide these polynomials using long division • 1. (4x2 – 9) ÷ (2x + 3) • 2. • 3. x4 – 1 • x2– 1

  24. What’s your identity task

  25. Factoring cubic binomials

  26. inverses

  27. homework • Worksheet

  28. Warm up! 4/17/2017 • Can you figure out what numbers would make the next row on this diagram?

  29. The binomial theorem • A binomial is a polynomial with two terms such as x + a. Often we need to raise a binomial to a power. In this section we'll explore a way to do just that without lengthy multiplication.

  30. Pascal’s triangle Can you see a pattern? Can you make a guess what the next one would be? What about the coefficients of the terms in the middle?

  31. + + + + + + + + + + Pascal’s triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 • Let's list all of the coefficients on the x's and the a's and look for a pattern. Can you guess the next row? 1 5 10 10 5 1

  32. Pascal’s triangle • This is good for lower powers but could get very large. There is notation to help us with the coefficients and a formula! • This is called Pascal's Triangle and would give us the coefficients for a binomial expansion of any power if we extended it far enough.

  33. ! ! The Factorial Symbol 0! = 1 1! = 1n! = n(n-1) · . . . · 3 · 2 · 1 n must be an integer greater than or equal to 2 • What this says is if you have a positive integer followed by the factorial symbol you multiply the integer by each integer less than it until you get down to 1. 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720 Your calculator can compute factorials. The ! symbol is under the "math" menu and then "prob". ! !

  34. combinations This symbol is read "n taken j at a time" Your calculator can compute these as well. It is also under the "math" and then "prob" menu and is usually denoted nCrwith the C meaning combinations. In probability, there are n things to choose from and you are choosing j of them for various combinations.

  35. Examples: 2 We are now ready to see how this applies to expanding binomials.

  36. The Binomial Theorem The x's start out to the nth power and decrease by 1 in power each term. The a’s start out to the 0 power and increase by 1 in power each term. The binomial coefficients are found by computing the combination symbol. Also the sum of the powers on a and x is n. Find the 5th term of (x + a)12 5th term will have a4(power on a is 1 less than term number) So we'll have x8(sum of two powers is 12) 1 less than term number

  37. Here is the expansion of (x + a)12 …and the 5th term matches the term we obtained! In this expansion, observe the following: • Powers on a and x add up to power of the binomial • a’s increase in power asx's decrease in power from term to term. • Powers on a are one less than the term number • symmetry of coefficients (i.e. 2nd term and 2nd to last term have same coefficients, 3rd & 3rd to last etc.) so once you've reached the middle, you can copy by symmetry rather than compute coefficients.

  38. Binomial expansion practice

  39. Warm up! 4/18/2017

  40. Binomial Expansion • The third term in the binomial expansion of (1+px)7 is 84x2. Find: a) The possible values of p • b) The fifth term in the expansion

  41. Expand (1+2/x)5

  42. Composition of functions worksheet • Please only complete the FRONT. The back of this worksheet if for our next learning goal, which we will start tomorrow.

  43. Warm UP 4/19/2017 • Factor the following: • 1) 27x3 + 8 • 2) x3y6 – 64 • 3) 16x3 – 250

  44. Unit 5 tests • Reassessment for Unit 5 is 4/26 after school • You must complete error analysis and the remediation provided on the blog. • Don’t forget: TODAY is the Unit 4 reassessment. If you have not already, please turn in the remediation and error analysis.

  45. Inverse applications • Standard maps of the earth are broken into a grid of latitude lines (east-west) and longitude lines (north-south). Consider the function, N(ℓ) , the percentage of earth's surface north of a given latitude, ℓ (north of the equator). Several values of N(ℓ) (to the nearest tenth) can be determined using the table below. • Use the data to sketch a graph of N(ℓ) for 30≤ℓ≤90 . • b. Is the graph of N(ℓ) increasing or decreasing? • c. What are the units of ℓ ? What are the units of N(ℓ) ? • d. What is the value of N−1(25) ? • e. Describe what is meant by the expression, N−1(20)

  46. Combining functions

  47. Combining functions • Let f be the function defined by f(x)=2x2+4x−16 . Let g be the function defined by g(x) =2(x+1)2−18. • Verify that f(x)=g(x) for all x . • In what ways do the equivalent expressions 2x2+4x−16 and 2(x+1)2−18 help to understand the function f ?

  48. Combining functions Let f be the function defined by f(x)=2x2 +4x−16 . Let g be the function defined by g(x) =2(x+1)2 −18. • Consider the functions h, l, m, and n given below. Show that f(x) is a composition, in some order, of the functions h, l, m, and n. How do you determine the order of composition? • Explain the impact each of the functions l, m, and n has on the graph of the composition.

  49. Equations and identities • Complete the 3 questions on the front. On the back, there is a series of equations. You should classify each equation as ALWAYS, SOMETIMES, or NEVER true.

  50. Warm UP 4/20/2017 Divide the following polynomials. Use which ever method is appropriate. 1) (6x³ + 17x² + 27x + 20) (3x + 4) • 2) • 3) • 4)

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