Exercice 1
Développer et réduire : \(A = (x + 4)^2\) ; \(B = (2x - 3)^2\) ; \(C = (5 - x)(5 + x)\) ; \(D = (3x + 1)(x - 2)\).
Correction
\(A = x^2 + 8x + 16\). \(B = 4x^2 - 12x + 9\). \(C = 25 - x^2\). \(D = 3x^2 - 6x + x - 2 = 3x^2 - 5x - 2\).
Exercice 2
Factoriser : \(E = 7x^2 - 14x\) ; \(F = x^2 - 10x + 25\) ; \(G = 9x^2 - 1\) ; \(H = 4x^2 + 12x + 9\).
Correction
\(E = 7x(x - 2)\). \(F = (x - 5)^2\). \(G = (3x - 1)(3x + 1)\). \(H = (2x + 3)^2\).
Exercice 3
Factoriser : \(I = (2x - 1)(x + 3) - (2x - 1)(4x - 5)\) et \(J = (x - 3)^2 - 16\).
Correction
\(I = (2x - 1)\left[(x + 3) - (4x - 5)\right] = (2x - 1)(-3x + 8)\). \(J = (x - 3 - 4)(x - 3 + 4) = (x - 7)(x + 1)\).
Exercice 4
Calculer sans calculatrice : \(99^2\) ; \(1002^2\) ; \(103\times97\).
Correction
\(99^2 = (100 - 1)^2 = 10\,000 - 200 + 1 = 9801\). \(1002^2 = 1\,000\,000 + 4000 + 4 = 1\,004\,004\). \(103\times97 = (100 + 3)(100 - 3) = 10\,000 - 9 = 9991\).
Exercice 5
On pose \(K = (x + 2)^2 - (x + 2)(3x - 4)\).
- Développer et réduire \(K\).
- Factoriser \(K\).
- Calculer \(K\) pour \(x = 3\).
Correction
- \(K = x^2 + 4x + 4 - (3x^2 - 4x + 6x - 8) = x^2 + 4x + 4 - 3x^2 - 2x + 8 = -2x^2 + 2x + 12\).
- \(K = (x + 2)\left[(x + 2) - (3x - 4)\right] = (x + 2)(-2x + 6)\).
- \(K = 5\times0 = 0\) (vérification : \(-18 + 6 + 12 = 0\)).