Notes on Rubik's Magic Cube by David Singmaster

By David Singmaster

Notes on Rubik's 'Magic dice'

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2 when x = 4. 1. EJ � 54. 25. = 55. Write a polynomial giving the area of a circle of radius r. 56. Write a polynomial giving the area of a triangle of base b and height h. S 57. 3. What does each of the following polynomials represent? 3 T 1 58. E. T. stock at $60 per share. What does the polynomial 55x + 45y + 60z represent? 2 ADDITION AND SUBTRACTION OF POLYNaVUALS Those terms of a polynomial which differ only in their coefficients are called like terms. Here are some examples of like terms: 4x 2 and -5xy3 and 2x 2y 2 and - 3x 2 17xy3 -2x 2y 2 We add polynomials by adding the coefficients of like terms.

3. 5. 7. 9. 1 1. 13. 15. 17. 19. 2+5=5+2 -2 · 5 = 5(-2) (4 + 3)2 = 4 . 2 + 3 . 2 -2(4 - 5) ( -2)(4) - 2( -5) -3 + (2 + 5) = ( - 3 + 2) + 5 3 +a=a+3 2(ab) (2a)b 2(xy) = x(2y) 5 + (a + 2) = 2 + (5 + a) = = (3 . 2)( -4) = 3[(2)( -4)] 2(4 + 5) = 2 . 4 + 2 . 5 3(2 - 4) = 3 . 2 - 3 . 4 (3 + 2) + 4 3 + (2 + 4) (2 -5) + 8 2 + ( -5 + 8) 2(x + 2) = 2x + 4 5(a + b) = 5(b + a) 4(a + b) 4b + 4a (5x)y = 5(yx) = = = Give examples showing that the operation of subtraction does not satisfy the commutative and associative laws.

2 Add the given polynomials in each of the following. _ l; l l. Find the mistake(s) in each of the following. Obtain the correct answer. = l = = Subtract the second polynomial from the first. l l. Subtract the first polynomial from the second. = 23. 25. 27. 29. 31. 33. 35. a2bc 3xy2z -4x2 ab2c 2ab3 -3a2bc -4ab3 3 $60 36. $50 $55x y $20 37. $260 x $13 z $17 x y 38. S + + + + + + + + + +s Simplify by combining like terms. T. stock at per share and y shares of Exxon stock at per share. E. stock at per share and shares of Bethlehem Steel stock at per share.

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