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Govt. of Tamilnadu Department of School Education Bridge Course 2011-2012 Class IX- maths

Govt. of Tamilnadu Department of School Education Bridge Course 2011-2012 Class IX- maths. Mathematicians. Pythagoras. 569 B.C. – 475 B.C. Greece First pure mathematician 5 beliefs Secret society Pythagorean theorem. Aristotle. 384 B.C. – 322 B.C. Greece Philosopher

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Govt. of Tamilnadu Department of School Education Bridge Course 2011-2012 Class IX- maths

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  1. Govt. of TamilnaduDepartment of School EducationBridge Course 2011-2012Class IX- maths

  2. Mathematicians

  3. Pythagoras • 569 B.C. – 475 B.C. • Greece • First pure mathematician • 5 beliefs • Secret society • Pythagorean theorem

  4. Aristotle • 384 B.C. – 322 B.C. • Greece • Philosopher • Studied mathematics in relation to science

  5. Euclid • 325 B.C. – 265 B.C. • Greece • Wrote The Elements • Geometry today

  6. Al-Khwarizmi • 780 A.D.-850 A.D. • Baghdad (in Iraq) • 1st book on Algebra • Algebra • Natural Number • Equation

  7. Algebra • Equation – A mathematical sentence stating that 2 expressions are equal. • 12 – 3 = 9 • 8 + 4 = 12

  8. Definitions • Equation – A mathematical sentence with an equals sign. • 16 – 5 = 11 • 14 + 3 = 17

  9. Definitions • Equals Sign (=) Means that the amount is the same on both sides. • 4 + 2 = 6 • 5 – 2 = 3

  10. Are these equations true, false or open? • 11 - 3 = 5 • 13 + 4 = 17 • N + 4 = 7 • 12 – 3 = 8 • 3 + v = 13 • 15 – 6 = 9 false true open false open true

  11. Definitions • Inverse operation – the opposite operation used to undo the first. • 4 + 3 = 7 7 – 4 = 3 • 6 x 6 = 36 36 / 6 = 6

  12. How to solve an addition equation • Use the inverse operation for addition which is subtraction • m+ 8 = 12 12 - 8 = 4 • m = 4 4 + 8 = 12

  13. How to solve a subtraction equation • Use the inverse operation for subtraction which is addition • m- 3 = 5 5 + 3 = 8 • m = 8 8 - 3 = 5

  14. Solve these equations using the inverse operations • n + 4 = 7 • n – 5 = 4 • n + 4 = 17 • n – 6 = 13 • n + 7 = 15 • n – 8 = 17 3 9 13 19 8 9

  15. Commutative Property • 5 + 4 = 9 4 + 5 = 9 • a + b = c b + a = c • 6 + 3 = 9 3 + 6 = 9 • x+ y = z y + x = z • 3 + 4 + 1 = 8 1 + 3 + 4 = 8

  16. Solve these equations using the commutative property • n + 7 = 7 + 4 • m + 2 = 2 + 5 • z + 3 = 3 + 9 • g + 6 = 6 + 11 • s + 4 = 4 + 20 • c + 8 = 8 + 32 n = 4 m = 5 z = 9 g = 11 s = 20 c = 32

  17. The Identity Property of Addition • 7 + 0 = 7 • a + 0 = a • 8 + 0 = 8 • c + 0 = c • 2 + 0 = 2

  18. Use the Identity Property of addition to solve these problems • n + 0 = 8 • b + 0 = 7 • m + 0 = 3 • v + 0 = 5 • w + 0 = 4 • r + 0 = 2 n = 8 b = 7 m = 3 v = 5 w = 4 r = 2

  19. Subtraction Rules of zero • 7 – 7 = 0 • n – n = 0 • 4 – 0 = 4 • n – 0 = n

  20. Find the value of n using the rules of subtraction • n - 8 = 0 • n – 9 = 0 • n – 0 = 7 • n – 0 = 9 • n – 7 = 0 • n – 0 = 5 n = 8 n = 9 n = 7 n = 9 n = 7 n = 5

  21. Polynomials What does each prefix mean? Mono One Bi Two Tri Three

  22. What about poly? one or more A polynomial is a monomial or a sum/difference of monomials. Important Note!! An expression is not a polynomial if there is a variable in the denominator.

  23. State whether each expression is a polynomial. If it is, identify it. 1) 7y - 3x + 4 trinomial 2) 10x3yz2 monomial 3) not a polynomial

  24. The degree of a monomial is the sum of the exponents of the variables.Find the degree of each monomial. 5x2 4a4b3c 3) 7x3Y5z

  25. To find the degree of a polynomial, find the largest degree of the terms. 1) 8x2 - 2x + 7 Degrees: 2 1 0 Which is biggest? 2 is the degree! 2) y7 + 6y4 + 3x4m4 Degrees: 7 4 8 8 is the degree!

  26. Find the degree of x5 – x3y2 + 4 • 0 • 2 • 3 • 5 • 10

  27. Put in descending order: • 8x - 3x2 + x4 - 4 x4 - 3x2 + 8x - 4 2) Put in descending order in terms of x: 12x2y3 - 6x3y2 + 3y - 2x -6x3y2 + 12x2y3 - 2x + 3y

  28. 3) Put in ascending order in terms of y: 12x2y3 - 6x3y2 + 3y - 2x -2x + 3y - 6x3y2 + 12x2y3 • Put in ascending order: 5a3 - 3 + 2a - a2 -3 + 2a - a2 + 5a3

  29. Write in ascending order in terms of y:x4 – x3y2 + 4xy–2x2y3 • x4 + 4xy– x3y2–2x2y3 • –2x2y3 – x3y2 + 4xy + x4 • x4 – x3y2–2x2y3 + 4xy • 4xy –2x2y3 – x3y2 + x4

  30. Addition And subtraction on polynomials1. Add the following polynomials:(9y - 7x + 15a) + (-3y + 8x - 8a) Group your like terms. 9y - 3y - 7x + 8x + 15a - 8a 6y + x + 7a

  31. 2. Add the following polynomials:(3a2 + 3ab - b2) + (4ab + 6b2) Combine your like terms. 3a2 + 3ab + 4ab - b2 + 6b2 3a2 + 7ab + 5b2

  32. Add the polynomials.+ Y X X2 Y X XY Y X Y 1 1 Y Y • x2 + 3x + 7y + xy + 8 • x2 + 4y + 2x + 3 • 3x + 7y + 8 • x2 + 11xy + 8 1 1 1 1 1 1 Y

  33. 3. Add the following polynomials using column form:(4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2) Line up your like terms. 4x2 - 2xy + 3y2 + -3x2 - xy + 2y2 _________________________ x2 - 3xy + 5y2

  34. Rewrite subtraction as adding the opposite. (9y - 7x + 15a) + (+ 3y - 8x + 8a) Group the like terms. 9y + 3y - 7x - 8x + 15a + 8a 12y - 15x + 23a 4. Subtract the following polynomials:(9y - 7x + 15a) - (-3y + 8x - 8a)

  35. 5. Subtract the following polynomials:(7a - 10b) - (3a + 4b) Rewrite subtraction as adding the opposite. (7a - 10b) + (- 3a - 4b) Group the like terms. 7a - 3a - 10b - 4b 4a - 14b

  36. 6. Subtract the following polynomials using column form:(4x2 - 2xy + 3y2) - (-3x2 - xy + 2y2) Line up your like terms and add the opposite. 4x2 - 2xy + 3y2 + (+ 3x2+ xy - 2y2) -------------------------------------- 7x2 - xy + y2

  37. Find the sum or difference.(5a – 3b) + (2a + 6b) • 3a – 9b • 3a + 3b • 7a + 3b • 7a – 3b

  38. Find the sum or difference.(5a – 3b) – (2a + 6b) • 3a – 9b • 3a + 3b • 7a + 3b • 7a – 9b

  39. (6 3)(y3 y5)   (3 9)(m m2)(n2 n)   Multiplying Monomials Multiply. A. (6y3)(3y5) (6y3)(3y5) Group factors with like bases together. Multiply. 18y8 B. (3mn2) (9m2n) Group factors with like bases together. (3mn2)(9m2n) 27m3n3 Multiply.

  40. 1 æ ö ( ) ( ) 2 2 2 s t t - 12 t s s ç ÷ 4 è ø 1 æ ö ( ) ( ) g g g g 2 2 2 - 12 s s s t t t g ÷ ç 4 ø è Multiplying Monomials Multiply. Group factors with like bases together. Multiply.

  41. (3 6)(x3 x2)   (2 5)(r2)(t3 t)   Example 1 Multiply. a. (3x3)(6x2) Group factors with like bases together. (3x3)(6x2) Multiply. 18x5 b. (2r2t)(5t3) Group factors with like bases together. (2r2t)(5t3) Multiply. 10r2t4

  42. Multiplying a Polynomial by a Monomial Multiply. 4(3x2 + 4x – 8) 4(3x2 + 4x – 8) Distribute 4. (4)3x2 +(4)4x – (4)8 Multiply. 12x2 + 16x – 32

  43. To multiply a binomial by a binomial, you can apply the Distributive Property more than once: Distribute x and 3. (x + 3)(x + 2) = x(x + 2)+ 3(x + 2) Distribute x and 3 again. = x(x + 2) + 3(x + 2) = x(x) + x(2) + 3(x) + 3(2) Multiply. = x2 + 2x + 3x + 6 Combine like terms. = x2 + 5x + 6

  44. 1. Multiply the First terms. (x + 3)(x + 2) x x = x2 2. Multiply the Outer terms. (x+ 3)(x+ 2) x 2 = 2x  3. Multiply the Inner terms. (x+ 3)(x+ 2) 3x = 3x  4. Multiply the Last terms. (x+ 3)(x+ 2) 3 2 = 6  (x + 3)(x + 2) = x2+2x + 3x +6 = x2 + 5x + 6 F O I L Another method for multiplying binomials is called the FOIL method. F O I L

  45. Multiplying Binomials Multiply. (s + 4)(s – 2) (s + 4)(s – 2) s(s – 2) + 4(s – 2) Distribute s and 4. s(s) + s(–2) + 4(s) + 4(–2) Distribute s and 4 again. s2 – 2s + 4s –8 Multiply. s2 + 2s –8 Combine like terms.

  46. (x x) + (x (–4)) + (–4  x) + (–4  (–4))   Multiplying Binomials Multiply. Write as a product of two binomials. (x – 4)2 (x – 4)(x – 4) Use the FOIL method. x2– 4x– 4x+8 Multiply. x2 – 8x + 8 Combine like terms.

  47. 3x4 12x3 –21x –7 x3 4x2 Multiplying Polynomials Multiply. (3x + 1)(x3 – 4x2 – 7) Write the product of the monomials in each row and column. –7 x3 4x2 3x Add all terms inside the rectangle. +1 3x4+ 12x3 + x3+ 4x2 – 21x – 7 3x4+ 13x3+ 4x2 – 21x – 7 Combine like terms.

  48. quotient + Dividing Polynomials Dividing Polynomials Long division of polynomials is similar to long division of whole numbers. When you divide two polynomials you can check the answer using the following: dividend = (quotient • divisor) + remainder The result is written in the form:

  49. 1. 2. 3. 4. 5. 6. –4 Answer: x + 2 + quotient (x + 2) (x + 1) + (– 4) = x2 + 3x – 2 divisor remainder dividend Example: Divide & Check Example: Divide x2 + 3x – 2 by x – 1 and check the answer. x + 2 x2+ x 2x – 2 2x + 2 –4 remainder Check: correct

  50. 1. 2. 3. 5 Answer: x2 + x +3 4. 5. 6. 9. 8. 7. Example: Divide & Check Example: Divide 4x + 2x3 – 1 by 2x – 2 and check the answer. + x + 3 x2 Write the terms of the dividend in descending order. 2x3 – 2x2 Since there is no x2 term in the dividend, add 0x2 as a placeholder. + 4x 2x2 2x2 – 2x 6x – 1 6x – 6 5 Check: (x2 + x + 3)(2x – 2) + 5 =4x + 2x3 – 1

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