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Understanding Polynomials: A Historical and Practical Perspective on Algebra

Explore the fascinating history and practical applications of polynomials, rooted in the ancient Arabic text "Hisab al-jabr w’al-muqabalah," which translates to "The science of equations." This guide delves into combining like terms, simplifying expressions, and practicing operations like addition and subtraction of polynomials. Learn techniques such as the distributive property and the FOIL method for multiplying binomials. Whether you're preparing for an exam or brushing up on your algebra skills, this comprehensive overview will enhance your understanding of polynomial operations.

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Understanding Polynomials: A Historical and Practical Perspective on Algebra

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  1. 1-4 Polynomial Review

  2. This is the title of a 300 year old book (in Arabic). Translated, it means: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” Does that part of the title look familiar?

  3. Consider going to aBuffalo Bills Game

  4. When you return from your trip, in your wallet you find some Canadian Money and some American money How much Canadian money do you have if you have a Loonie, a $5 and a $20? 1 (Canadian) + 5 (Canadian) + 20 (Canadian) = 1C + 5C + 20C = 26C You can add this money together because it is the same currency

  5. Suppose you have $20.00 American and $26.00 CanadianWhat is your total amount? 20 (American) + 26 (Canadian) = ? $46 Ca-merican?not so much… 20A + 26C = 20A + 26C The money can not be combined because they are of different currencies (they are not like terms…)

  6. “Like Terms” contain the same variable-exponent Only “Like Terms” can be combined x and x are like termsx3and x3are like termsx andx3are not like terms

  7. Simplify: 3x + 7y – 2x = 1x + 7y -2a + 7b + 4a + 10a = 12a + 7b

  8. Add (4x2 + 2x + 5) + (2x2 – 3x + 4) = 6x2 - 1x + 9

  9. Subtract (6x2 + 1x + 9) - (3x2 – 3x + 5) = 3x2 +4x + 4

  10. Operations with Polynomials BEDMAS suggests that all brackets must be eliminated first when operating. To do this, apply the 1.DISTRIBUTIVE PROPERTY

  11. In general: a(b + c) = ab + ac

  12. For example: 2x(4x + 2y) = 8x2 + 4xy

  13. For example: 3x(7a + 6b) = 21ax +18bx

  14. 3(2a + 7b) + 2(a + 5b) = 6a + 21b +2a + 10b = 8a + 31b

  15. 3(2a + 7b) - 2(a + 5b) = 6a + 21b -2a -10b = 4a + 11b

  16. 6k(3 – k + 2k2) = 18k – 6k2 + 12k3 5s(3s2 –2s -7) = 15s3 – 10s2 - 35s

  17. To multiply two binomials, you must use:2. FOIL

  18. F O I L (a + b)(c + d) = ac + ad + bc + bd FIRST OUTSIDE INSIDE LAST

  19. For example: (2x + 7)(4x + 2) = 8x2 + 4x +28x +14 Combine the middle two like terms = 8x2 + 32x + 14

  20. Expand and Simplify 2(x + 3) + (x + 1)(2x – 4) + 3x(x – 7) Use the Distributive Property on these Use FOIL on this

  21. Expand and Simplify 2(x + 3) + (x + 1)(2x – 4) + 3x(x – 7) = 2x + 6 + 2x2 – 4x + 2x - 4 + 3x2 – 21x = 5x2 -21x + 2

  22. Page 29 (1,2) a,c 7 Page 31 {8,9,10} ace Page 33 [11,12,13] ace

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