1. **Énoncé du problème :** Calculer les expressions algébriques données avec les binômes $A = x + 2$, $B = x + 7$, $C = x^2 - 3$, $D = 3x + 1$. 2. **Rappel des règles importantes :** - Pour additionner des binômes, on additionne terme à terme. - Pour multiplier, on utilise la distributivité : $(a+b)(c+d) = ac + ad + bc + bd$. - Pour élever au carré un binôme, on utilise la formule $(a+b)^2 = a^2 + 2ab + b^2$. 3. **Calculs :** **a) $A + B$** $$A + B = (x + 2) + (x + 7) = x + x + 2 + 7 = 2x + 9$$ **b) $B + D$** $$B + D = (x + 7) + (3x + 1) = x + 3x + 7 + 1 = 4x + 8$$ **c) $AD$** $$AD = (x + 2)(3x + 1) = x \times 3x + x \times 1 + 2 \times 3x + 2 \times 1 = 3x^2 + x + 6x + 2 = 3x^2 + 7x + 2$$ **d) $A(B + C)$** $$B + C = (x + 7) + (x^2 - 3) = x^2 + x + 4$$ $$A(B + C) = (x + 2)(x^2 + x + 4) = x \times (x^2 + x + 4) + 2 \times (x^2 + x + 4) = x^3 + x^2 + 4x + 2x^2 + 2x + 8 = x^3 + 3x^2 + 6x + 8$$ **e) $(A + B)^2$** $$A + B = 2x + 9$$ $$ (A + B)^2 = (2x + 9)^2 = (2x)^2 + 2 \times 2x \times 9 + 9^2 = 4x^2 + 36x + 81$$ **f) $A + C$** $$A + C = (x + 2) + (x^2 - 3) = x^2 + x - 1$$ **g) $C + D$** $$C + D = (x^2 - 3) + (3x + 1) = x^2 + 3x - 2$$ **h) $AB + AC$** $$AB = (x + 2)(x + 7) = x^2 + 7x + 2x + 14 = x^2 + 9x + 14$$ $$AC = (x + 2)(x^2 - 3) = x^3 - 3x + 2x^2 - 6 = x^3 + 2x^2 - 3x - 6$$ $$AB + AC = (x^2 + 9x + 14) + (x^3 + 2x^2 - 3x - 6) = x^3 + 3x^2 + 6x + 8$$ **i) $A^2 + 2AB + B^2$** $$A^2 = (x + 2)^2 = x^2 + 4x + 4$$ $$B^2 = (x + 7)^2 = x^2 + 14x + 49$$ $$2AB = 2 \times (x + 2)(x + 7) = 2(x^2 + 9x + 14) = 2x^2 + 18x + 28$$ $$A^2 + 2AB + B^2 = (x^2 + 4x + 4) + (2x^2 + 18x + 28) + (x^2 + 14x + 49) = 4x^2 + 36x + 81$$ **j) $A + D$** $$A + D = (x + 2) + (3x + 1) = 4x + 3$$ **k) $AB$** $$AB = (x + 2)(x + 7) = x^2 + 9x + 14$$ **l) $BD$** $$BD = (x + 7)(3x + 1) = 3x^2 + x + 21x + 7 = 3x^2 + 22x + 7$$ **m) $D(A + B + C)$** $$A + B + C = (x + 2) + (x + 7) + (x^2 - 3) = x^2 + 2x + 6$$ $$D(A + B + C) = (3x + 1)(x^2 + 2x + 6) = 3x \times x^2 + 3x \times 2x + 3x \times 6 + 1 \times x^2 + 1 \times 2x + 1 \times 6 = 3x^3 + 6x^2 + 18x + x^2 + 2x + 6 = 3x^3 + 7x^2 + 20x + 6$$ **n) $B + C$** $$B + C = (x + 7) + (x^2 - 3) = x^2 + x + 4$$ **o) $AC$** $$AC = (x + 2)(x^2 - 3) = x^3 + 2x^2 - 3x - 6$$ **p) $CD$** $$CD = (x^2 - 3)(3x + 1) = x^2 \times 3x + x^2 \times 1 - 3 \times 3x - 3 \times 1 = 3x^3 + x^2 - 9x - 3$$