exercice 1

Factoriser H(x) = x² - x - 2
aide : calculer H(2)

exercice 2

Factoriser R(x) = x³ + 2x² - 5x - 6
aide :
calculer R(2)
a³ - b³ = (a - b)(a² + ab + b²)

exercice 3

Factoriser A(x) = 2x² + 10x - 3

exercice 1

H(x) = x² - x - 2
Calculons H(2) : H(2) = 2² - 2 - 2 = 0
Donc : H(x) = H(x) - H(2)
H(x) = x² - x - 2 - (2² - 2 - 2)
H(x) = (x² - 2²) - (x - 2)
H(x) = (x - 2)(x + 2) - (x - 2)
H(x) = (x - 2)(x + 2 - 1)
H(x) = (x - 2)(x + 1)

exercice 2

R(x) = x³ + 2x² - 5x - 6
Calculons R(2) = 2³ + 2 × 2² - 5 × 2 - 6 = 0
Donc : R(x) = R(x) - R(2)
R(x) = x³ + 2x² - 5x - 6 - (2³ + 2 × 2² - 5 × 2 - 6)
R(x) = (x³ - 2³) + 2(x² - 2²) - 5(x - 2)
R(x) = (x - 2)(x² + 2x + 4) + 2(x - 2)(x + 2) - 5(x - 2)
R(x) = (x - 2)(x² + 2x+ 4 + 2x + 4 - 5)
R(x) = (x - 2)(x² + 4x + 3)

Pour aller plus loin...
On prend :
Z(x) = x² + 4x + 3
0 = Z(-1) = (-1)² + 4 × (-1) + 3
____________________________
Z(x) = Z(x) - Z(-1) = (x² - (-1)²) + 4(x - (-1))
Z(x) = (x - 1)(x + 1) + 4(x + 1)
Z(x) = (x + 1)(x + 3)

Donc pour reprendre R(x) :
R(x) = x³ + 2x - 5x - 6
R(x) = (x - 2)(x² + 4x + 3)
R(x) = (x - 2)(x + 1)(x + 3)

exercice 3

On va utiliser la forme canonique ...