ARITMETICA E ALGEBRA PRACTICE WITH CLIL Square numbers Square numbers are natural numbers that can be represented as squares. For example, 22 and 32 are square numbers. 32 may be reproduced in the following pattern: Which is the relationship between what is observed and the general formula for the square of the binomial? It is quite easy to understand that to find the next square number, you add to the given pattern a dot in the corner (the blue one in the figure) plus - on both sides - a number of dots equivalent to the previous squared number. Fig. 1 Now, consider a generic natural number n (which square number is n2). Write in symbols its next number raised to power 2 and develop the square of the binomial resulting from the operation. You will have: (n + 1)2 = n2 + 2n + 1 Therefore, in figure 2, you will find: 32 + 1 + 2 3 = 9 + 1 + 6 = 16 = 42 and, its next square number will be: 42 + 1 + 2 4 = 16 + 1 + 8 = 25 = 52 In general, the square number next to n2 can be represented as follows: (n + 1)2 = n2 + 1 + 2n which is the development of the binomial square n + 1. Now, observe the figure 2 showing how to create the square of the next of 3 starting from 32: Fig. 2 VERSO LA PROVA DI VERIFICA CONOSCENZE ABILIT Riconosci le tecniche utilizzate per effettuate le seguenti scomposizioni di polinomi. Scomponi in fattori i seguenti polinomi. 4 2 3 2 5. 4 a4 8 a3 b a2 b4 + 4 a2 b2 + 2a b5 b6 2 1. x y + x y x y = x y(x + 1 x) 6. x2 18x + 45 2. 9 a2 + 12 + 8b + 6 a2 b = (3 a2 + 4)(3 + 2b) 2 1 1 1 7. a2 ab + _ ac + _ b2 _ bc + _ c2 3 4 3 9 3. 9 x5 12 x3 y + 4x y2 = x (3 x2 2y)2 4. x4 16 y4 = (x + 2y)(x 2y)(x2 + 4 y2) 8. 1000 p3 + 729 q3 9. Determina il mcm e il MCD dei seguenti polinomi. 3x2 x 3xy + y; 9x2 6x + 1 3xy 3y2; PROBLEM SOLVING 10. Senza sviluppare il quadrato del binomio dimostra che (x + p)2 x2 è multiplo di p. AUTOVALUTAZIONE Indica con una crocetta gli esercizi che hai risolto in modo corretto. Esercizi 396 1 2 3 4 5 6 7 8 9 10