1 / 46

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics. §5.1 Intro to PolyNomials. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. MTH 55. 4.3. Review §. Any QUESTIONS About §4.3 → Absolute Value: Equations & InEqualities Any QUESTIONS About HomeWork §4.3 → HW-14. Mathematical “TERMS”.

chauncey-arturo
Download Presentation

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chabot Mathematics §5.1 Intro toPolyNomials Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. MTH 55 4.3 Review § • Any QUESTIONS About • §4.3 → Absolute Value: Equations & InEqualities • Any QUESTIONS About HomeWork • §4.3 → HW-14

  3. Mathematical “TERMS” • A TERM can be a number, a variable, a product of numbers and/or variables, or a quotient of numbers and/or variables. • A term that is a product of constants and/or variables is called a monomial. • Examples of monomials: 8, w, 24x3y • A polynomial is a monomial or a sum of monomials. Examples of polynomials: • 5w + 8, −3x2 + x + 4, x, 0, 75y6

  4. Example  Terms • Identify the terms of the polynomial 7p5 − 3p3 + 3 • SOLUTION • The terms are 7p5, −3p3, and 3. • We can see this by rewriting all subtractions as additions of opposites: 7p5− 3p3 + 3 = 7p5 + (−3p3) + 3 These are the terms of the polynomial.

  5. [Bi, Tri, Poly]-nomials • A polynomial that is composed of two terms is called a binomial, whereas those composed of three terms are called trinomials. Polynomials with four or more terms have no special name

  6. Polynomial DEGREE • The degree of a term of a polynomial is the no. of variable factors in that term • EXAMPLE: Determine the degree of each term: a) 9x5 b) 6y c) 9 • SOLUTION • a) The degree of 9x5 is 5 • b) The degree of 6y (6y1) is 1 • c) The degree of 9 (9z0) is 0

  7. Mathematical COEFFICIENT • The part of a term that is a constantfactor is the coefficient of that term. The coefficient of 4y is 4. • EXAMPLE: Identify the coefficient of each term in polynomial: 5x4− 8x2y + y− 9 • SOLUTION • The coefficient of 5x4 is 5. • The coefficient of −8x2y is −8. • The coefficient of y is 1, since y = 1y. • The coefficient of −9 is simply −9

  8. DEGREE of POLYNOMIAL • The leading term of a polynomial is the term of highest degree. Its coefficient is called the leading coefficient and its degree is referred to as the degree of the polynomial. • Consider this polynomial 4x2– 9x3 + 6x4 + 8x– 7. • Find the TERMS, COEFFICIENTS, and DEGREE

  9. DEGREE of POLYNOMIAL • For polynomial: 4x2− 9x3 + 6x4 + 8x− 7 • List Terms, CoEfficients, Term-Degree • Terms →4x2, −9x3,6x4, 8x, and−7 • coefficients → 4, −9, 6, 8 and −7 • degree of each term → 2, 3, 4, 1, and 0 • The leading term is 6x4 and the leading coefficientis 6. • The degree of the polynomialis 4.

  10. Example  −3x4 + 6x3− 2x2 + 8x + 7 • Complete Table for PolyNomial–3x4 + 6x3 – 2x2 + 8x + 7

  11. Example  –3x4 + 6x3 – 2x2 + 8x + 7 • Put Terms in Descending Exponent Order

  12. Example  –3x4 + 6x3 – 2x2 + 8x + 7 • Coefficients are the CONSTANTS before the Variables

  13. Example  –3x4 + 6x3 – 2x2 + 8x + 7 • Term DEGREE is the Value of the EXPONENT

  14. Example  –3x4 + 6x3 – 2x2 + 8x + 7 • Polymomial Degree is the SAME as the highest Term Degree

  15. MultiVariable PolyNomials • Evaluate the 2-Var polynomial 5 + 4x + xy2 + 9x3y2 for x = −3 & y = 4 • Solution: Substitute −3 for x and 4 for y: 5 + 4x + xy2 + 9x3y2 = 5 + 4(−3) + (−3)(4)2 + 9(−3)3(4)2 = 5 − 12 − 48 − 3888 = −3943

  16. Degree of MultiVar Polynomial • Recall that the degree of a polynomial is the number of variable factors in the term. • Example: ID the coefficient and the degree of each term and the degree of the polynomial 10x3y2– 15xy3z4 + yz + 5y + 3x2 + 9

  17. Like Terms • Like, or similarterms either have exactly the same variables with exactly the same exponents or are constants. • For example, 9w5y4 and 15w5y4 are like terms • and −12 and 14 are like terms, • but −6x2y and 9xy3 are not like terms.

  18. Example  Combine Like Terms • 10x2y + 4xy3− 6x2y − 2xy3 • 8st− 6st2 + 4st2 + 7s3 + 10st − 12s3 + t − 2 • SOLUTION • 10x2y + 4xy3− 6x2y− 2xy3 = (10 − 6)x2y + (4 − 2)xy3 = 4x2y + 2xy3 • 8st− 6st2 + 4st2 + 7s3 + 10st− 12s3+ t− 2 = −5s3− 2st2 + 18st + t − 2

  19. Common Properties: PolyNom Fcns • The domain of a polynomial function is the set of all real numbers. • The graph of a polynomial function is a continuous curve. • This means that the graph has no holes or gaps and can be drawn on a piece of paper without lifting the pencil.

  20. Continuous vs. DisContinuous Could be a PolyNomial Can NOT be a PolyNomial

  21. Common Properties: PolyNom Fcns • The graph of a polynomial function is a smooth curve. • This means that the graph of a polynomial function does NOT contain any SHARP corners.

  22. Smooth vs. Kinked/Cornered Can NOT be a PolyNomial Could be a PolyNomial

  23. Leading Coefficient Test • Given a PolyNomial Function of the form • The leading term is anxn. The behavior of the graph of f(x) as x →  or as x → −is dominated by this term, and is similar to one of the following 4 graphs • Note that The middle portion of each graph, indicated by the dashed lines, is NOT determined by this test.

  24. Lead Coeff Test: Odd & Positive • Leading Term • ODD Exponent • POSITIVE Coeff • Graph • FALLS to LEFT • RISES to RIGHT

  25. Lead Coeff Test: Odd & Negative • Leading Term • ODD Exponent • NEGATIVE Coeff • Graph • RISES to LEFT • FALLS to RIGHT

  26. Lead Coeff Test: Even & Positive • Leading Term • EVEN Exponent • POSITIVE Coeff • Graph • RISES to LEFT • RISES to RIGHT

  27. Lead Coeff Test: Even & Negative • Leading Term • EVEN Exponent • NEGATIVE Coeff • Graph • FALLS to LEFT • FALLS to RIGHT

  28. Example  Lead CoEff Test • Use the leading-CoEfficient test to determine the end behavior of the graph of • SOLUTION • Here n = 3 (odd) and an = −2 < 0. Thus, Case-2 (Odd & Neg) applies. The graph of f(x) rises to the left and falls to the right. This behavior is described by: y → as x → −;and y → −as x → 

  29. Adding Polynomials • EXAMPLE  Add (−6x3 + 7x− 2) + (5x3 + 4x2 + 3) • Solution → Combine Like terms (−6x3 + 7x− 2) + (5x3 + 4x2 + 3) = (−6 + 5)x3+ 4x2 + 7x + (−2 + 3) = −x3 + 4x2 + 7x + 1

  30. Example  Add Polynomials • Add: (3 – 4x + 2x2) + (–6 + 8x– 4x2 + 2x3) • Solution(3 –4x + 2x2) + (–6 + 8x–4x2 + 2x3) = (3 – 6) + (–4 + 8)x + (2–4)x2 + 2x3 = –3 + 4x– 2x2 + 2x3

  31. Example  Add Polynomials • Add: 10x5– 3x3 + 7x2 + 4 and 6x4– 8x2 + 7 and 4x6– 6x5 + 2x2 + 6 • Solution 10x5 - 3x3 + 7x2 + 4 6x4 - 8x2 + 7 4x6 - 6x5 + 2x2 + 6 4x6 + 4x5 + 6x4 - 3x3 + x2 + 17 • Answer: 4x6 + 4x5 + 6x4− 3x3 + x2 + 17

  32. Opposite of a PolyNomial • To find an equivalent polynomial for the opposite, or additive inverse, of a polynomial, change the sign of every term. • This is the same as multiplying the original polynomial by −1.

  33. Example  Opposite of PolyNom • Simplify: –(–8x4–x3 + 9x2– 2x + 72) • Solution –(–8x4–x3 + 9x2– 2x + 72) = (–1)(–8x4–x3 + 9x2– 2x + 72) = 8x4 + x3– 9x2 + 2x– 72

  34. PolyNomial Subtraction • We can now subtract one polynomial from another by adding the opposite of the polynomial being subtracted. PolyNomial Subtractor

  35. Example  Subtract PolyNom • (10x5 + 2x3– 3x2 + 5) – (–3x5 + 2x4– 5x3– 4x2) • Solution (10x5 + 2x3– 3x2 + 5) – (–3x5 + 2x4– 5x3– 4x2) = 10x5 + 2x3 – 3x2 + 5 +3x5– 2x4 + 5x3 + 4x2 = 13x5 – 2x4 + 7x3 + x2 + 5

  36. Example  Subtract • (8x5 + 2x3– 10x) – (4x5– 5x3 + 6) • Solution (8x5 + 2x3– 10x) – (4x5– 5x3 + 6) =8x5 + 2x3– 10x+(–4x5) + 5x3– 6 = 4x5 + 7x3– 10x– 6

  37. Example  Column Form • Write in columns and subtract: (6x2– 4x + 7) – (10x2– 6x– 4) • Solution 6x2– 4x + 7 –(10x2– 6x– 4) –4x2 + 2x + 11 Remember to Change the SIGNS

  38. WhiteBoard Work • Problems From §5.1 Exercise Set • By ppt → 22, 24, 26, 28, 70 • 10 Adding and Subtracting Functions If f(x) and g(x) define functions, then (f + g) (x) = f (x) + g(x) Sum function and (f – g) (x) = f (x) – g(x). Difference function In each case, the domain of the new function is theintersection of the domains of f(x) and g(x).

  39. P5.1-[22, 24] • PolyNomial orNOT PolyNomial KINKED → NOT a Polynomial SMOOTH & CONTINUOUS → IS a Polynomial

  40. P5.1-[26, 28] • Use Lead CoEfficient Test of End Behavior to Match Fcn to Graph Odd & Negs → Rise-Lt & Falls-Rt Odd & Pos → Falls-Lt & Rises-Rt

  41. P5.1-70  AIDS Mortality Models • Given PolyNomial Models for USA AIDS mortality over the years 1990-2002 where x≡ yrs since 1990 • Bar Chart shows ACTUAL 2002 Mortality of 501 669 • Find Error Associated with Each Model

  42. Evaluate Model using MATLAB Math-Processing Software See MTH25 for Info on MATLAB P5.1-70  AIDS Mortality Models >> x =2002-1990 x = 12 >> fx = -1844*x^2 + 54923*x + 111568 fx = 505108 >> gx = -11*x^3 - 2066*x^2 + 56036*x + 110590 gx = 466510 >> Yactual = 501669 >> fx_error = (fx-Yactual)/Yactual fx_error = 0.0069 = 0.69% >> gx_error = (gx-Yactual)/Yactual gx_error = -0.0701 = -7.01% • By MATLAB the Model Errors • f(x) → 0.69% Low • g(x) → 7.0% Low

  43. n is Even an > 0 n is Even an < 0 n is Odd an > 0 n is Odd an < 0 All Done for Today Lead CoeffTestSummarized

  44. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

  45. Graph y = |x| • Make T-table

More Related