Factorisation par facteur commun| Exercices corrigés | 4e

Objectif : maîtriser la factorisation par mise en évidence d’un facteur commun grâce à des exercices corrigés et progressifs.

Exercice 1

Repérer le facteur commun, puis factoriser les expressions suivantes :

$A \> =2x+2y\\$
$B \> =8a+8b\\$
$C \> =5x+y\times5\\$
$D \> =a\times4+b\times4$

$E \> =2x+y\times2+2z\\$
$F \> =a\times7+7b+c\times7\\$
$G \> =12x+y\times12+12z\\$
$H \> =a\times10+b\times10+10c$

Afficher ou masquer la correction

$\begin{aligned}
A &= 2x+2y\\
&= 2(x+y)
\end{aligned}$

$\begin{aligned}
B &= 8a+8b\\
&= 8(a+b)
\end{aligned}$

$\begin{aligned}
C &= 5x+y\times5\\
&= 5x+5y\\
&= 5(x+y)
\end{aligned}$

$\begin{aligned}
D &= a\times4+b\times4\\
&= 4a+4b\\
&= 4(a+b)
\end{aligned}$

$\begin{aligned}
E &= 2x+y\times2+2z\\
&= 2x+2y+2z\\
&= 2(x+y+z)
\end{aligned}$

$\begin{aligned}
F &= a\times7+7b+c\times7\\
&= 7a+7b+7c\\
&= 7(a+b+c)
\end{aligned}$

$\begin{aligned}
G &= 12x+y\times12+12z\\
&= 12x+12y+12z\\
&= 12(x+y+z)
\end{aligned}$

$\begin{aligned}
H &= a\times10+b\times10+10c\\
&= 10a+10b+10c\\
&= 10(a+b+c)
\end{aligned}$

Exercice 2

Repérer le facteur commun, puis factoriser les expressions suivantes :

$A \> =3x-3y\\$
$B \> =6a-6b\\$
$C \> =7x-y\times7\\$
$D \> =a\times9-b\times9$

$E \> =4x-y\times4+4z\\$
$F \> =a\times11+11b-c\times11\\$
$G \> =6x+y\times6-6z\\$
$H \> =a\times8-b\times8-8c$

Afficher ou masquer la correction

$\begin{aligned}
A &= 3x-3y\\
&= 3(x-y)
\end{aligned}$

$\begin{aligned}
B &= 6a-6b\\
&= 6(a-b)
\end{aligned}$

$\begin{aligned}
C &= 7x-y\times7\\
&= 7x-7y\\
&= 7(x-y)
\end{aligned}$

$\begin{aligned}
D &= a\times9-b\times9\\
&= 9a-9b\\
&= 9(a-b)
\end{aligned}$

$\begin{aligned}
E &= 4x-y\times4+4z\\
&= 4x-4y+4z\\
&= 4(x-y+z)
\end{aligned}$

$\begin{aligned}
F &= a\times11+11b-c\times11\\
&= 11a+11b-11c\\
&= 11(a+b-c)
\end{aligned}$

$\begin{aligned}
G &= 6x+y\times6-6z\\
&= 6x+6y-6z\\
&= 6(x+y-z)
\end{aligned}$

$\begin{aligned}
H &= a\times8-b\times8-8c\\
&= 8a-8b-8c\\
&= 8(a-b-c)
\end{aligned}$

Exercice 3

Repérer le plus grand facteur commun, puis factoriser au maximum les expressions suivantes :

$A \> =5a+20\\$
$B \> =14x-21\\$
$C \> =18a+24\\$
$D \> =35x-15$

$E \> =8x+12y\\$
$F \> =27a-36b\\$
$G \> =16x+24y-40\\$
$H \> =21a-14b+35$

$I \> =12x-18y\\$
$J \> =45a+30b\\$
$K \> =24x+36y+60\\$
$L \> =32a-48b+16c$

$M \> =18x+30y-42\\$
$N \> =25a-40b+15\\$
$O \> =28x-42y-70\\$
$P \> =54a+72b-90c$

Afficher ou masquer la correction

$\begin{aligned}
A &= 5a+20\\
&= 5a+5\times4\\
&= 5(a+4)
\end{aligned}$

$\begin{aligned}
B &= 14x-21\\
&= 7\times2x-7\times3\\
&= 7(2x-3)
\end{aligned}$

$\begin{aligned}
C &= 18a+24\\
&= 6\times3a+6\times4\\
&= 6(3a+4)
\end{aligned}$

$\begin{aligned}
D &= 35x-15\\
&= 5\times7x-5\times3\\
&= 5(7x-3)
\end{aligned}$

$\begin{aligned}
E &= 8x+12y\\
&= 4\times2x+4\times3y\\
&= 4(2x+3y)
\end{aligned}$

$\begin{aligned}
F &= 27a-36b\\
&= 9\times3a-9\times4b\\
&= 9(3a-4b)
\end{aligned}$

$\begin{aligned}
G &= 16x+24y-40\\
&= 8\times2x+8\times3y-8\times5\\
&= 8(2x+3y-5)
\end{aligned}$

$\begin{aligned}
H &= 21a-14b+35\\
&= 7\times3a-7\times2b+7\times5\\
&= 7(3a-2b+5)
\end{aligned}$

$\begin{aligned}
I &= 12x-18y\\
&= 6\times2x-6\times3y\\
&= 6(2x-3y)
\end{aligned}$

$\begin{aligned}
J &= 45a+30b\\
&= 15\times3a+15\times2b\\
&= 15(3a+2b)
\end{aligned}$

$\begin{aligned}
K &= 24x+36y+60\\
&= 12\times2x+12\times3y+12\times5\\
&= 12(2x+3y+5)
\end{aligned}$

$\begin{aligned}
L &= 32a-48b+16c\\
&= 16\times2a-16\times3b+16\times c\\
&= 16(2a-3b+c)
\end{aligned}$

$\begin{aligned}
M &= 18x+30y-42\\
&= 6\times3x+6\times5y-6\times7\\
&= 6(3x+5y-7)
\end{aligned}$

$\begin{aligned}
N &= 25a-40b+15\\
&= 5\times5a-5\times8b+5\times3\\
&= 5(5a-8b+3)
\end{aligned}$

$\begin{aligned}
O &= 28x-42y-70\\
&= 14\times2x-14\times3y-14\times5\\
&= 14(2x-3y-5)
\end{aligned}$

$\begin{aligned}
P &= 54a+72b-90c\\
&= 18\times3a+18\times4b-18\times5c\\
&= 18(3a+4b-5c)
\end{aligned}$

Exercice 4

Repérer le plus grand facteur commun, puis factoriser au maximum les expressions suivantes :

$A \> =12x+6\\$
$B \> =5-20a\\$
$C \> =9+18x\\$
$D \> =24a-8$

$E \> =5-35x\\$
$F \> =42a+7\\$
$G \> =8-32x\\$
$H \> =15+45a$

Afficher ou masquer la correction

$\begin{aligned}
A &= 12x+6\\
&= 6\times2x+6\times1\\
&= 6(2x+1)
\end{aligned}$

$\begin{aligned}
B &= 5-20a\\
&= 5\times1-5\times4a\\
&= 5(1-4a)
\end{aligned}$

$\begin{aligned}
C &= 9+18x\\
&= 9\times1+9\times2x\\
&= 9(1+2x)
\end{aligned}$

$\begin{aligned}
D &= 24a-8\\
&= 8\times3a-8\times1\\
&= 8(3a-1)
\end{aligned}$

$\begin{aligned}
E &= 5-35x\\
&= 5\times1-5\times7x\\
&= 5(1-7x)
\end{aligned}$

$\begin{aligned}
F &= 42a+7\\
&= 7\times6a+7\times1\\
&= 7(6a+1)
\end{aligned}$

$\begin{aligned}
G &= 8-32x\\
&= 8\times1-8\times4x\\
&= 8(1-4x)
\end{aligned}$

$\begin{aligned}
H &= 15+45a\\
&= 15\times1+15\times3a\\
&= 15(1+3a)
\end{aligned}$

Exercice 5

Repérer le plus grand facteur commun, puis factoriser chaque expression par ce facteur :

$A \> =3x+x^2\\$
$B \> =x^2+5x\\$
$C \> =4x^2+8x\\$
$D \> =6x^2-9x$

$E \> =x^3+4x^2\\$
$F \> =5x^3-10x^2\\$
$G \> =12x^3+18x^2\\$
$H \> =15x^4-10x^3$

$I \> =x^4+3x^2+2x\\$
$J \> =4x^5+8x^3-12x\\$
$K \> =6x^5-9x^4+3x^2\\$
$L \> =10x^6+15x^3-20x^2$

$M \> =14x^7-21x^5+7x^3\\$
$N \> =18x^6+12x^4-6x^2\\$
$O \> =24x^8-36x^5+12x^4\\$
$P \> =30x^9+45x^6-15x^3$

Afficher ou masquer la correction

$\begin{aligned}
A &= 3x+x^2\\
&= x\times3+x\times x\\
&= x(3+x)
\end{aligned}$

$\begin{aligned}
B &= x^2+5x\\
&= x\times x+x\times5\\
&= x(x+5)
\end{aligned}$

$\begin{aligned}
C &= 4x^2+8x\\
&= 4x\times x+4x\times2\\
&= 4x(x+2)
\end{aligned}$

$\begin{aligned}
D &= 6x^2-9x\\
&= 3x\times2x-3x\times3\\
&= 3x(2x-3)
\end{aligned}$

$\begin{aligned}
E &= x^3+4x^2\\
&= x^2\times x+x^2\times4\\
&= x^2(x+4)
\end{aligned}$

$\begin{aligned}
F &= 5x^3-10x^2\\
&= 5x^2\times x-5x^2\times2\\
&= 5x^2(x-2)
\end{aligned}$

$\begin{aligned}
G &= 12x^3+18x^2\\
&= 6x^2\times2x+6x^2\times3\\
&= 6x^2(2x+3)
\end{aligned}$

$\begin{aligned}
H &= 15x^4-10x^3\\
&= 5x^3\times3x-5x^3\times2\\
&= 5x^3(3x-2)
\end{aligned}$

$\begin{aligned}
I &= x^4+3x^2+2x\\
&= x\times x^3+x\times3x+x\times2\\
&= x(x^3+3x+2)
\end{aligned}$

$\begin{aligned}
J &= 4x^5+8x^3-12x\\
&= 4x\times x^4+4x\times2x^2-4x\times3\\
&= 4x(x^4+2x^2-3)
\end{aligned}$

$\begin{aligned}
K &= 6x^5-9x^4+3x^2\\
&= 3x^2\times2x^3-3x^2\times3x^2+3x^2\times1\\
&= 3x^2(2x^3-3x^2+1)
\end{aligned}$

$\begin{aligned}
L &= 10x^6+15x^3-20x^2\\
&= 5x^2\times2x^4+5x^2\times3x-5x^2\times4\\
&= 5x^2(2x^4+3x-4)
\end{aligned}$

$\begin{aligned}
M &= 14x^7-21x^5+7x^3\\
&= 7x^3\times2x^4-7x^3\times3x^2+7x^3\times1\\
&= 7x^3(2x^4-3x^2+1)
\end{aligned}$

$\begin{aligned}
M &= 14x^7-21x^5+7x^3\\
&= 7x^3\times2x^4-7x^3\times3x^2\\[0.5ex]
&\quad{}+7x^3\times1\\
&= 7x^3(2x^4-3x^2+1)
\end{aligned}$

$\begin{aligned}
N &= 18x^6+12x^4-6x^2\\
&= 6x^2\times3x^4+6x^2\times2x^2-6x^2\times1\\
&= 6x^2(3x^4+2x^2-1)
\end{aligned}$

$\begin{aligned}
N &= 18x^6+12x^4-6x^2\\
&= 6x^2\times3x^4+6x^2\times2x^2\\[0.5ex]
&\quad{}-6x^2\times1\\
&= 6x^2(3x^4+2x^2-1)
\end{aligned}$

$\begin{aligned}
O &= 24x^8-36x^5+12x^4\\
&= 12x^4\times2x^4-12x^4\times3x+12x^4\times1\\
&= 12x^4(2x^4-3x+1)
\end{aligned}$

$\begin{aligned}
O &= 24x^8-36x^5+12x^4\\
&= 12x^4\times2x^4-12x^4\times3x\\[0.5ex]
&\quad{}+12x^4\times1\\
&= 12x^4(2x^4-3x+1)
\end{aligned}$

$\begin{aligned}
P &= 30x^9+45x^6-15x^3\\
&= 15x^3\times2x^6+15x^3\times3x^3-15x^3\times1\\
&= 15x^3(2x^6+3x^3-1)
\end{aligned}$

$\begin{aligned}
P &= 30x^9+45x^6-15x^3\\
&= 15x^3\times2x^6+15x^3\times3x^3\\[0.5ex]
&\quad{}-15x^3\times1\\
&= 15x^3(2x^6+3x^3-1)
\end{aligned}$

Exercice 6

Repérer le plus grand facteur commun à tous les termes, puis factoriser chaque expression par ce facteur :

$A \> =3(x+1)+x(x+1)\\$
$B \> =(x+2)\times5+(x+2)\times x\\$
$C \> =2(x-3)+(x-3)\times x\\$
$D \> =x(x+4)+7(x+4)$

$E \> =(3x-2)\times xy-4z(3x-2)\\$
$F \> =3z(2x+1)+(2x+1)\times xyz\\$
$G \> =2xy(x+2)+(x+2)\times xyz\\$
$H \> =(2x+y)\times xyz-5xy(2x+y)$

Afficher ou masquer la correction

$\begin{aligned}
A &= 3(x+1)+x(x+1)\\
&= (x+1)\times3+(x+1)\times x\\
&= (x+1)(3+x)
\end{aligned}$

$\begin{aligned}
B &= (x+2)\times5+(x+2)\times x\\
&= (x+2)(5+x)
\end{aligned}$

$\begin{aligned}
C &= 2(x-3)+(x-3)\times x\\
&= (x-3)\times2+(x-3)\times x\\
&= (x-3)(2+x)
\end{aligned}$

$\begin{aligned}
D &= x(x+4)+7(x+4)\\
&= (x+4)\times x+(x+4)\times7\\
&= (x+4)(x+7)
\end{aligned}$

$\begin{aligned}
E &= (3x-2)\times xy-4z(3x-2)\\
&= (3x-2)\times xy-(3x-2)\times4z\\
&= (3x-2)(xy-4z)
\end{aligned}$

$\begin{aligned}
F &= 3z(2x+1)+(2x+1)\times xyz\\
&= z(2x+1)\times3+z(2x+1)\times xy\\
&= z(2x+1)(3+xy)
\end{aligned}$

$\begin{aligned}
G &= 2xy(x+2)+(x+2)\times xyz\\
&= xy(x+2)\times2+xy(x+2)\times z\\
&= xy(x+2)(2+z)
\end{aligned}$

$\begin{aligned}
H &= (2x+y)\times xyz-5xy(2x+y)\\
&= xy(2x+y)\times z-xy(2x+y)\times5\\
&= xy(2x+y)(z-5)
\end{aligned}$

Exercice 7

Factoriser les expressions suivantes :

$A \> =5(x-2)+(3x+1)(x-2)\\$
$B \> =(2x+3)(x+4)-7(x+4)\\$
$C \> =(x-5)(3x+2)+4(3x+2)\\$
$D \> =6(2x-3)-(x+1)(2x-3)$

$E \> =(3x+4)(2x-1)+(x-5)(2x-1)\\$
$F \> =(2x+7)(3x-2)-(x+4)(3x-2)\\$
$G \> =(x^2+7)(x+3)+5x(x+3)\\$
$H \> =(2x^2-x+3)(3x-4)-(x^2+2)(3x-4)$

Afficher ou masquer la correction

$\begin{aligned}
A &= 5(x-2)+(3x+1)(x-2)\\
&= (x-2)\times5+(x-2)\times(3x+1)\\
&= (x-2)[5+(3x+1)]\\
&= (x-2)(5+3x+1)\\
&= (x-2)(3x+6)\\
&= (x-2)(3\times x+3\times2)\\
&= (x-2)[3(x+2)]\\
&= (x-2)\times3\times(x+2)\\
&= 3(x-2)(x+2)
\end{aligned}$

$\begin{aligned}
B &= (2x+3)(x+4)-7(x+4)\\
&= (x+4)\times(2x+3)-(x+4)\times7\\
&= (x+4)[(2x+3)-7]\\
&= (x+4)(2x+3-7)\\
&= (x+4)(2x-4)\\
&= (x+4)(2\times x-2\times2)\\
&= (x+4)[2(x-2)]\\
&= (x+4)\times2\times(x-2)\\
&= 2(x+4)(x-2)
\end{aligned}$

$\begin{aligned}
C &= (x-5)(3x+2)+4(3x+2)\\
&= (3x+2)\times(x-5)+(3x+2)\times4\\
&= (3x+2)[(x-5)+4]\\
&= (3x+2)(x-5+4)\\
&= (3x+2)(x-1)
\end{aligned}$

$\begin{aligned}
D &= 6(2x-3)-(x+1)(2x-3)\\
&= (2x-3)\times6-(2x-3)\times(x+1)\\
&= (2x-3)[6-(x+1)]\\
&= (2x-3)(6-x-1)\\
&= (2x-3)(5-x)
\end{aligned}$

$\begin{aligned}
E &= (3x+4)(2x-1)+(x-5)(2x-1)\\
&= (2x-1)\times(3x+4)+(2x-1)\times(x-5)\\
&= (2x-1)[(3x+4)+(x-5)]\\
&= (2x-1)(3x+4+x-5)\\
&= (2x-1)(4x-1)
\end{aligned}$

$\begin{aligned}
E &= (3x+4)(2x-1)\\[0.5ex]
&\quad{}+(x-5)(2x-1)\\[1ex]
&= (2x-1)\times(3x+4)\\[0.5ex]
&\quad{}+(2x-1)\times(x-5)\\[1ex]
&= (2x-1)[(3x+4)+(x-5)]\\
&= (2x-1)(3x+4+x-5)\\
&= (2x-1)(4x-1)
\end{aligned}$

$\begin{aligned}
F &= (2x+7)(3x-2)-(x+4)(3x-2)\\
&= (3x-2)\times(2x+7)-(3x-2)\times(x+4)\\
&= (3x-2)[(2x+7)-(x+4)]\\
&= (3x-2)(2x+7-x-4)\\
&= (3x-2)(x+3)
\end{aligned}$

$\begin{aligned}
F &= (2x+7)(3x-2)-(x+4)(3x-2)\\[1ex]
&= (3x-2)\times(2x+7)\\[0.5ex]
&\quad{}-(3x-2)\times(x+4)\\[1ex]
&= (3x-2)[(2x+7)-(x+4)]\\
&= (3x-2)(2x+7-x-4)\\
&= (3x-2)(x+3)
\end{aligned}$

$\begin{aligned}
G &= (x^2+7)(x+3)+5x(x+3)\\
&= (x+3)\times(x^2+7)+(x+3)\times5x\\
&= (x+3)[(x^2+7)+5x]\\
&= (x+3)(x^2+7+5x)\\
&= (x+3)(x^2+5x+7)
\end{aligned}$

$\begin{aligned}
H &= (2x^2-x+3)(3x-4)-(x^2+2)(3x-4)\\
&= (3x-4)\times(2x^2-x+3)-(3x-4)\times(x^2+2)\\
&= (3x-4)[(2x^2-x+3)-(x^2+2)]\\
&= (3x-4)(2x^2-x+3-x^2-2)\\
&= (3x-4)(x^2-x+1)
\end{aligned}$

$\begin{aligned}
H &= (2x^2-x+3)(3x-4)\\[0.5ex]
&\quad{}-(x^2+2)(3x-4)\\[1ex]
&= (3x-4)\times(2x^2-x+3)\\[0.5ex]
&\quad{}-(3x-4)\times(x^2+2)\\[1ex]
&= (3x-4)[(2x^2-x+3)-(x^2+2)]\\
&= (3x-4)(2x^2-x+3-x^2-2)\\
&= (3x-4)(x^2-x+1)
\end{aligned}$

Exercice 8

Factoriser les expressions suivantes :

$A \> =2xyz(x+3)+(x+3)\times4z-3z(x+3)\\$
$B \> =(2x-1)\times xy+4xyz(2x-1)-(2x-1)\times2xy\\$
$C \> =2xy(x+2)+(x+2)\times10xyz-6xy(x+2)\\$
$D \> =3(2x+y)\times xyz-12x^{2}yz(2x+y)+(2x+y)\times6xyz^{2}$

$E \> =18x^2y-12xy^2+30xy\\$
$F \> =8a^4b^2+12a^3b^3+16a^5b\\$
$G \> =24x^3y^2+18x^2y^3+30x^4y\\$
$H \> =14a^5b^2+21a^3b^4-28a^4b^3$

Afficher ou masquer la correction

$\begin{aligned}
A &= 2xyz(x+3)+(x+3)\times4z-3z(x+3)\\
&= z(x+3)\times2xy+z(x+3)\times4-z(x+3)\times3\\
&= z(x+3)(2xy+4-3)\\
&= z(x+3)(2xy+1)
\end{aligned}$

$\begin{aligned}
A &= 2xyz(x+3)+(x+3)\times4z\\[0.5ex]
&\quad{}-3z(x+3)\\[1ex]
&= z(x+3)\times2xy+z(x+3)\times4\\[0.5ex]
&\quad{}-z(x+3)\times3\\[1ex]
&= z(x+3)(2xy+4-3)\\
&= z(x+3)(2xy+1)
\end{aligned}$

$\begin{aligned}
B &= (2x-1)\times xy+4xyz(2x-1)-(2x-1)\times2xy\\
&= xy(2x-1)\times1+xy(2x-1)\times4z-xy(2x-1)\times2\\
&= xy(2x-1)(1+4z-2)\\
&= xy(2x-1)(4z-1)
\end{aligned}$

$\begin{aligned}
B &= (2x-1)\times xy+4xyz(2x-1)\\[0.5ex]
&\quad{}-(2x-1)\times2xy\\[1ex]
&= xy(2x-1)\times1+xy(2x-1)\times4z\\[0.5ex]
&\quad{}-xy(2x-1)\times2\\[1ex]
&= xy(2x-1)(1+4z-2)\\
&= xy(2x-1)(4z-1)
\end{aligned}$

$\begin{aligned}
C &= 2xy(x+2)+(x+2)\times10xyz-6xy(x+2)\\
&= 2xy(x+2)\times1+2xy(x+2)\times5z-2xy(x+2)\times3\\
&= 2xy(x+2)(1+5z-3)\\
&= 2xy(x+2)(5z-2)
\end{aligned}$

$\begin{aligned}
C &= 2xy(x+2)+(x+2)\times10xyz\\[0.5ex]
&\quad{}-6xy(x+2)\\[1ex]
&= 2xy(x+2)\times1+2xy(x+2)\times5z\\[0.5ex]
&\quad{}-2xy(x+2)\times3\\[1ex]
&= 2xy(x+2)(1+5z-3)\\
&= 2xy(x+2)(5z-2)
\end{aligned}$

$\begin{aligned}
D &= 3(2x+y)\times xyz-12x^2yz(2x+y)+(2x+y)\times6xyz^2\\
&= 3xyz(2x+y)\times1-3xyz(2x+y)\times4x+3xyz(2x+y)\times2z\\
&= 3xyz(2x+y)(1-4x+2z)
\end{aligned}$

$\begin{aligned}
D &= 3(2x+y)\times xyz\\[0.5ex]
&\quad{}-12x^2yz(2x+y)\\[0.5ex]
&\quad{}+(2x+y)\times6xyz^2\\[1ex]
&= 3xyz(2x+y)\times1\\[0.5ex]
&\quad{}-3xyz(2x+y)\times4x\\[0.5ex]
&\quad{}+3xyz(2x+y)\times2z\\[1ex]
&= 3xyz(2x+y)(1-4x+2z)
\end{aligned}$

$\begin{aligned}
E &= 18x^2y-12xy^2+30xy\\
&= 6xy\times3x-6xy\times2y+6xy\times5\\
&= 6xy(3x-2y+5)
\end{aligned}$

$\begin{aligned}
F &= 8a^4b^2+12a^3b^3+16a^5b\\
&= 4a^3b\times2ab+4a^3b\times3b^2+4a^3b\times4a^2\\
&= 4a^3b(2ab+3b^2+4a^2)
\end{aligned}$

$\begin{aligned}
F &= 8a^4b^2+12a^3b^3+16a^5b\\
&= 4a^3b\times2ab+4a^3b\times3b^2\\[0.5ex]
&\quad{}+4a^3b\times4a^2\\[1ex]
&= 4a^3b(2ab+3b^2+4a^2)
\end{aligned}$

$\begin{aligned}
G &= 24x^3y^2+18x^2y^3+30x^4y\\
&= 6x^2y\times4xy+6x^2y\times3y^2+6x^2y\times5x^2\\
&= 6x^2y(4xy+3y^2+5x^2)
\end{aligned}$

$\begin{aligned}
G &= 24x^3y^2+18x^2y^3+30x^4y\\
&= 6x^2y\times4xy+6x^2y\times3y^2\\[0.5ex]
&\quad{}+6x^2y\times5x^2\\[1ex]
&= 6x^2y(4xy+3y^2+5x^2)
\end{aligned}$

$\begin{aligned}
H &= 14a^5b^2+21a^3b^4-28a^4b^3\\
&= 7a^3b^2\times2a^2+7a^3b^2\times3b^2-7a^3b^2\times4ab\\
&= 7a^3b^2(2a^2+3b^2-4ab)
\end{aligned}$

$\begin{aligned}
H &= 14a^5b^2+21a^3b^4-28a^4b^3\\
&= 7a^3b^2\times2a^2+7a^3b^2\times3b^2\\[0.5ex]
&\quad{}-7a^3b^2\times4ab\\[1ex]
&= 7a^3b^2(2a^2+3b^2-4ab)
\end{aligned}$

Exercice 9

Factoriser les expressions suivantes :

$A \> =(3x-7)(2x+5)+6x-14\\$
$B \> =4x(5x+1)-15x-3\\$
$C \> =10x+25-(2x+5)(x-6)\\$
$D \> =(x-2)(5x+4)+6-3x$

$E \> =(x+4)(7-2x)+(2x-7)(3x-1)\\$
$F \> =(2x+1)(x-5)-4x-2+6x+3\\$
$G \> =(4x-3)^2-(8x-6)(x+2)\\$
$H \> =(3x-2)(x+4)+(2-3x)(2x-1)+9x-6$

Afficher ou masquer la correction

$\begin{aligned}
A &= (3x-7)(2x+5)+6x-14\\
&= (3x-7)(2x+5)+2\times3x-2\times7\\
&= (3x-7)(2x+5)+2(3x-7)\\
&= (3x-7)\times(2x+5)+(3x-7)\times2\\
&= (3x-7)[(2x+5)+2]\\
&= (3x-7)(2x+5+2)\\
&= (3x-7)(2x+7)
\end{aligned}$

$\begin{aligned}
B &= 4x(5x+1)-15x-3\\
&= 4x(5x+1)-(15x+3)\\
&= 4x(5x+1)-(3\times5x+3\times1)\\
&= 4x(5x+1)-[3(5x+1)]\\
&= 4x(5x+1)-3(5x+1)\\
&= (5x+1)\times4x-(5x+1)\times3\\
&= (5x+1)(4x-3)
\end{aligned}$

$\begin{aligned}
C &= 10x+25-(2x+5)(x-6)\\
&= 5\times2x+5\times5-(2x+5)(x-6)\\
&= 5(2x+5)-(2x+5)(x-6)\\
&= (2x+5)\times5-(2x+5)\times(x-6)\\
&= (2x+5)[5-(x-6)]\\
&= (2x+5)(5-x+6)\\
&= (2x+5)(11-x)
\end{aligned}$

$\begin{aligned}
D &= (x-2)(5x+4)+6-3x\\
&= (x-2)(5x+4)-3x+6\\
&= (x-2)(5x+4)-(3x-6)\\
&= (x-2)(5x+4)-(3\times x-3\times2)\\
&= (x-2)(5x+4)-[3(x-2)]\\
&= (x-2)(5x+4)-3(x-2)\\
&= (x-2)\times(5x+4)-(x-2)\times3\\
&= (x-2)[(5x+4)-3]\\
&= (x-2)(5x+4-3)\\
&= (x-2)(5x+1)
\end{aligned}$

$\begin{aligned}
E &= (x+4)(7-2x)+(2x-7)(3x-1)\\
&= (x+4)(7-2x)+[-(7-2x)](3x-1)\\
&= (x+4)(7-2x)-(7-2x)(3x-1)\\
&= (7-2x)\times(x+4)-(7-2x)\times(3x-1)\\
&= (7-2x)[(x+4)-(3x-1)]\\
&= (7-2x)(x+4-3x+1)\\
&= (7-2x)(5-2x)
\end{aligned}$

$\begin{aligned}
E &= (x+4)(7-2x)\\[0.5ex]
&\quad{}+(2x-7)(3x-1)\\[1ex]
&= (x+4)(7-2x)\\[0.5ex]
&\quad{}+[-(7-2x)](3x-1)\\[1ex]
&= (x+4)(7-2x)-(7-2x)(3x-1)\\
&= (7-2x)\times(x+4)\\[0.5ex]
&\quad{}-(7-2x)\times(3x-1)\\[1ex]
&= (7-2x)[(x+4)-(3x-1)]\\
&= (7-2x)(x+4-3x+1)\\
&= (7-2x)(5-2x)
\end{aligned}$

$\begin{aligned}
F &= (2x+1)(x-5)-4x-2+6x+3\\
&= (2x+1)(x-5)-(4x+2)+(6x+3)\\
&= (2x+1)(x-5)-(2\times2x+2\times1)+(3\times2x+3\times1)\\
&= (2x+1)(x-5)-[2(2x+1)]+[3(2x+1)]\\
&= (2x+1)(x-5)-2(2x+1)+3(2x+1)\\
&= (2x+1)\times(x-5)-(2x+1)\times2+(2x+1)\times3\\
&= (2x+1)[(x-5)-2+3]\\
&= (2x+1)(x-5-2+3)\\
&= (2x+1)(x-4)
\end{aligned}$

$\begin{aligned}
F &= (2x+1)(x-5)-4x-2\\[0.5ex]
&\quad{}+6x+3\\[1ex]
&= (2x+1)(x-5)-(4x+2)\\[0.5ex]
&\quad{}+(6x+3)\\[1ex]
&= (2x+1)(x-5)-(2\times2x+2\times1)\\[0.5ex]
&\quad{}+(3\times2x+3\times1)\\[1ex]
&= (2x+1)(x-5)-[2(2x+1)]\\[0.5ex]
&\quad{}+[3(2x+1)]\\[1ex]
&= (2x+1)(x-5)-2(2x+1)\\[0.5ex]
&\quad{}+3(2x+1)\\[1ex]
&= (2x+1)\times(x-5)-(2x+1)\times2\\[0.5ex]
&\quad{}+(2x+1)\times3\\[1ex]
&= (2x+1)[(x-5)-2+3]\\
&= (2x+1)(x-5-2+3)\\
&= (2x+1)(x-4)
\end{aligned}$

$\begin{aligned}
G &= (4x-3)^2-(8x-6)(x+2)\\
&= (4x-3)(4x-3)-(8x-6)(x+2)\\
&= (4x-3)(4x-3)-(2\times4x-2\times3)(x+2)\\
&= (4x-3)(4x-3)-[2(4x-3)](x+2)\\
&= (4x-3)(4x-3)-2(4x-3)(x+2)\\
&= (4x-3)\times(4x-3)-(4x-3)\times[2(x+2)]\\
&= (4x-3)[(4x-3)-2(x+2)]\\
&= (4x-3)(4x-3-2x-4)\\
&= (4x-3)(2x-7)
\end{aligned}$

$\begin{aligned}
G &= (4x-3)^2-(8x-6)(x+2)\\
&= (4x-3)(4x-3)-(8x-6)(x+2)\\
&= (4x-3)(4x-3)\\[0.5ex]
&\quad{}-(2\times4x-2\times3)(x+2)\\[1ex]
&= (4x-3)(4x-3)\\[0.5ex]
&\quad{}-[2(4x-3)](x+2)\\[1ex]
&= (4x-3)(4x-3)-2(4x-3)(x+2)\\
&= (4x-3)\times(4x-3)\\[0.5ex]
&\quad{}-(4x-3)\times[2(x+2)]\\[1ex]
&= (4x-3)[(4x-3)-2(x+2)]\\
&= (4x-3)(4x-3-2x-4)\\
&= (4x-3)(2x-7)
\end{aligned}$

$\begin{aligned}
H &= (3x-2)(x+4)+(2-3x)(2x-1)+9x-6\\
&= (3x-2)(x+4)+[-(3x-2)](2x-1)+3\times3x-3\times2\\
&= (3x-2)(x+4)-(3x-2)(2x-1)+3(3x-2)\\
&= (3x-2)\times(x+4)-(3x-2)\times(2x-1)+(3x-2)\times3\\
&= (3x-2)[(x+4)-(2x-1)+3]\\
&= (3x-2)(x+4-2x+1+3)\\
&= (3x-2)(8-x)
\end{aligned}$

$\begin{aligned}
H &= (3x-2)(x+4)+(2-3x)(2x-1)\\[0.5ex]
&\quad{}+9x-6\\[1ex]
&= (3x-2)(x+4)\\[0.5ex]
&\quad{}+[-(3x-2)](2x-1)\\[0.5ex]
&\quad{}+3\times3x-3\times2\\[1ex]
&= (3x-2)(x+4)-(3x-2)(2x-1)\\[0.5ex]
&\quad{}+3(3x-2)\\[1ex]
&= (3x-2)\times(x+4)\\[0.5ex]
&\quad{}-(3x-2)\times(2x-1)\\[0.5ex]
&\quad{}+(3x-2)\times3\\[1ex]
&= (3x-2)[(x+4)-(2x-1)+3]\\
&= (3x-2)(x+4-2x+1+3)\\
&= (3x-2)(8-x)
\end{aligned}$

Exercice 10

Factoriser les expressions suivantes :

$A \> =(7x-2)(x+3)+(2-7x)(2x-1)+14x-4\\$
$B \> =(3x+2)(2x-5)+9x+6-(3x+2)(x-1)\\$
$C \> =(4x-7)(3x+1)+8x-14-(7-4x)(x+2)\\$
$D \> =(5x+1)(x-4)+(4-x)(2x-3)+10x-40$

$E \> =(x-6)(5x+4)-(3x-18)+12-2x\\$
$F \> =(2x+5)^2-(x-1)(2x+5)-(8x+20)\\$
$G \> =(4x+3)(x-2)+(2x+1)(4x+3)-(12x+9)\\$
$H \> =(3x-4)(2x+7)+(4-3x)(x-5)+6x-8$

Afficher ou masquer la correction

$\begin{aligned}
A &= (7x-2)(x+3)+(2-7x)(2x-1)+14x-4\\
&= (7x-2)(x+3)+[-(7x-2)](2x-1)+2\times7x-2\times2\\
&= (7x-2)(x+3)-(7x-2)(2x-1)+2(7x-2)\\
&= (7x-2)\times(x+3)-(7x-2)\times(2x-1)+(7x-2)\times2\\
&= (7x-2)[(x+3)-(2x-1)+2]\\
&= (7x-2)(x+3-2x+1+2)\\
&= (7x-2)(6-x)
\end{aligned}$

$\begin{aligned}
A &= (7x-2)(x+3)\\[0.5ex]
&\quad{}+(2-7x)(2x-1)+14x-4\\[1ex]
&= (7x-2)(x+3)\\[0.5ex]
&\quad{}+[-(7x-2)](2x-1)\\[0.5ex]
&\quad{}+2\times7x-2\times2\\[1ex]
&= (7x-2)(x+3)\\[0.5ex]
&\quad{}-(7x-2)(2x-1)+2(7x-2)\\[1ex]
&= (7x-2)\times(x+3)\\[0.5ex]
&\quad{}-(7x-2)\times(2x-1)+(7x-2)\times2\\[1ex]
&= (7x-2)[(x+3)-(2x-1)+2]\\
&= (7x-2)(x+3-2x+1+2)\\
&= (7x-2)(6-x)
\end{aligned}$

$\begin{aligned}
B &= (3x+2)(2x-5)+9x+6-(3x+2)(x-1)\\
&= (3x+2)(2x-5)+3\times3x+3\times2-(3x+2)(x-1)\\
&= (3x+2)(2x-5)+3(3x+2)-(3x+2)(x-1)\\
&= (3x+2)\times(2x-5)+(3x+2)\times3-(3x+2)\times(x-1)\\
&= (3x+2)[(2x-5)+3-(x-1)]\\
&= (3x+2)(2x-5+3-x+1)\\
&= (3x+2)(x-1)
\end{aligned}$

$\begin{aligned}
B &= (3x+2)(2x-5)+9x+6\\[0.5ex]
&\quad{}-(3x+2)(x-1)\\[1ex]
&= (3x+2)(2x-5)+3\times3x+3\times2\\[0.5ex]
&\quad{}-(3x+2)(x-1)\\[1ex]
&= (3x+2)(2x-5)+3(3x+2)\\[0.5ex]
&\quad{}-(3x+2)(x-1)\\[1ex]
&= (3x+2)\times(2x-5)+(3x+2)\times3\\[0.5ex]
&\quad{}-(3x+2)\times(x-1)\\[1ex]
&= (3x+2)[(2x-5)+3-(x-1)]\\
&= (3x+2)(2x-5+3-x+1)\\
&= (3x+2)(x-1)
\end{aligned}$

$\begin{aligned}
C &= (4x-7)(3x+1)+8x-14-(7-4x)(x+2)\\
&= (4x-7)(3x+1)+2\times4x-2\times7-[-(4x-7)](x+2)\\
&= (4x-7)(3x+1)+2(4x-7)+(4x-7)(x+2)\\
&= (4x-7)\times(3x+1)+(4x-7)\times2+(4x-7)\times(x+2)\\
&= (4x-7)[(3x+1)+2+(x+2)]\\
&= (4x-7)(3x+1+2+x+2)\\
&= (4x-7)(4x+5)
\end{aligned}$

$\begin{aligned}
C &= (4x-7)(3x+1)+8x-14\\[0.5ex]
&\quad{}-(7-4x)(x+2)\\[1ex]
&= (4x-7)(3x+1)+2\times4x-2\times7\\[0.5ex]
&\quad{}-[-(4x-7)](x+2)\\[1ex]
&= (4x-7)(3x+1)+2(4x-7)\\[0.5ex]
&\quad{}+(4x-7)(x+2)\\[1ex]
&= (4x-7)\times(3x+1)+(4x-7)\times2\\[0.5ex]
&\quad{}+(4x-7)\times(x+2)\\[1ex]
&= (4x-7)[(3x+1)+2+(x+2)]\\
&= (4x-7)(3x+1+2+x+2)\\
&= (4x-7)(4x+5)
\end{aligned}$

$\begin{aligned}
D &= (5x+1)(x-4)+(4-x)(2x-3)+10x-40\\
&= (5x+1)(x-4)+[-(x-4)](2x-3)+10\times x-10\times4\\
&= (5x+1)(x-4)-(x-4)(2x-3)+10(x-4)\\
&= (x-4)\times(5x+1)-(x-4)\times(2x-3)+(x-4)\times10\\
&= (x-4)[(5x+1)-(2x-3)+10]\\
&= (x-4)(5x+1-2x+3+10)\\
&= (x-4)(3x+14)
\end{aligned}$

$\begin{aligned}
D &= (5x+1)(x-4)\\[0.5ex]
&\quad{}+(4-x)(2x-3)+10x-40\\[1ex]
&= (5x+1)(x-4)\\[0.5ex]
&\quad{}+[-(x-4)](2x-3)\\[0.5ex]
&\quad{}+10\times x-10\times4\\[1ex]
&= (5x+1)(x-4)\\[0.5ex]
&\quad{}-(x-4)(2x-3)+10(x-4)\\[1ex]
&= (x-4)\times(5x+1)\\[0.5ex]
&\quad{}-(x-4)\times(2x-3)\\[0.5ex]
&\quad{}+(x-4)\times10\\[1ex]
&= (x-4)[(5x+1)-(2x-3)+10]\\
&= (x-4)(5x+1-2x+3+10)\\
&= (x-4)(3x+14)
\end{aligned}$

$\begin{aligned}
E &= (x-6)(5x+4)-(3x-18)+12-2x\\
&= (x-6)(5x+4)-(3x-18)-(2x-12)\\
&= (x-6)(5x+4)-(3\times x-3\times6)-(2\times x-2\times6)\\
&= (x-6)(5x+4)-[3(x-6)]-[2(x-6)]\\
&= (x-6)(5x+4)-3(x-6)-2(x-6)\\
&= (x-6)\times(5x+4)-(x-6)\times3-(x-6)\times2\\
&= (x-6)[(5x+4)-3-2]\\
&= (x-6)(5x+4-3-2)\\
&= (x-6)(5x-1)
\end{aligned}$

$\begin{aligned}
E &= (x-6)(5x+4)-(3x-18)\\[0.5ex]
&\quad{}+12-2x\\[1ex]
&= (x-6)(5x+4)-(3x-18)\\[0.5ex]
&\quad{}-(2x-12)\\[1ex]
&= (x-6)(5x+4)-(3\times x-3\times6)\\[0.5ex]
&\quad{}-(2\times x-2\times6)\\[1ex]
&= (x-6)(5x+4)-[3(x-6)]\\[0.5ex]
&\quad{}-[2(x-6)]\\[1ex]
&= (x-6)(5x+4)-3(x-6)\\[0.5ex]
&\quad{}-2(x-6)\\[1ex]
&= (x-6)\times(5x+4)-(x-6)\times3\\[0.5ex]
&\quad{}-(x-6)\times2\\[1ex]
&= (x-6)[(5x+4)-3-2]\\
&= (x-6)(5x+4-3-2)\\
&= (x-6)(5x-1)
\end{aligned}$

$\begin{aligned}
F &= (2x+5)^2-(x-1)(2x+5)-(8x+20)\\
&= (2x+5)(2x+5)-(x-1)(2x+5)-(4\times2x+4\times5)\\
&= (2x+5)(2x+5)-(x-1)(2x+5)-[4(2x+5)]\\
&= (2x+5)(2x+5)-(x-1)(2x+5)-4(2x+5)\\
&= (2x+5)\times(2x+5)-(2x+5)\times(x-1)-(2x+5)\times4\\
&= (2x+5)[(2x+5)-(x-1)-4]\\
&= (2x+5)(2x+5-x+1-4)\\
&= (2x+5)(x+2)
\end{aligned}$

$\begin{aligned}
F &= (2x+5)^2\\[0.5ex]
&\quad{}-(x-1)(2x+5)-(8x+20)\\[1ex]
&= (2x+5)(2x+5)\\[0.5ex]
&\quad{}-(x-1)(2x+5)\\[0.5ex]
&\quad{}-(4\times2x+4\times5)\\[1ex]
&= (2x+5)(2x+5)\\[0.5ex]
&\quad{}-(x-1)(2x+5)-[4(2x+5)]\\[1ex]
&= (2x+5)(2x+5)\\[0.5ex]
&\quad{}-(x-1)(2x+5)-4(2x+5)\\[1ex]
&= (2x+5)\times(2x+5)\\[0.5ex]
&\quad{}-(2x+5)\times(x-1)\\[0.5ex]
&\quad{}-(2x+5)\times4\\[1ex]
&= (2x+5)[(2x+5)-(x-1)-4]\\
&= (2x+5)(2x+5-x+1-4)\\
&= (2x+5)(x+2)
\end{aligned}$

$\begin{aligned}
G &= (4x+3)(x-2)+(2x+1)(4x+3)-(12x+9)\\
&= (4x+3)(x-2)+(2x+1)(4x+3)-(3\times4x+3\times3)\\
&= (4x+3)(x-2)+(2x+1)(4x+3)-[3(4x+3)]\\
&= (4x+3)(x-2)+(2x+1)(4x+3)-3(4x+3)\\
&= (4x+3)\times(x-2)+(4x+3)\times(2x+1)-(4x+3)\times3\\
&= (4x+3)[(x-2)+(2x+1)-3]\\
&= (4x+3)(x-2+2x+1-3)\\
&= (4x+3)(3x-4)
\end{aligned}$

$\begin{aligned}
G &= (4x+3)(x-2)\\[0.5ex]
&\quad{}+(2x+1)(4x+3)-(12x+9)\\[1ex]
&= (4x+3)(x-2)\\[0.5ex]
&\quad{}+(2x+1)(4x+3)\\[0.5ex]
&\quad{}-(3\times4x+3\times3)\\[1ex]
&= (4x+3)(x-2)\\[0.5ex]
&\quad{}+(2x+1)(4x+3)-[3(4x+3)]\\[1ex]
&= (4x+3)(x-2)\\[0.5ex]
&\quad{}+(2x+1)(4x+3)-3(4x+3)\\[1ex]
&= (4x+3)\times(x-2)\\[0.5ex]
&\quad{}+(4x+3)\times(2x+1)\\[0.5ex]
&\quad{}-(4x+3)\times3\\[1ex]
&= (4x+3)[(x-2)+(2x+1)-3]\\
&= (4x+3)(x-2+2x+1-3)\\
&= (4x+3)(3x-4)
\end{aligned}$

$\begin{aligned}
H &= (3x-4)(2x+7)+(4-3x)(x-5)+6x-8\\
&= (3x-4)(2x+7)+[-(3x-4)](x-5)+2\times3x-2\times4\\
&= (3x-4)(2x+7)-(3x-4)(x-5)+2(3x-4)\\
&= (3x-4)\times(2x+7)-(3x-4)\times(x-5)+(3x-4)\times2\\
&= (3x-4)[(2x+7)-(x-5)+2]\\
&= (3x-4)(2x+7-x+5+2)\\
&= (3x-4)(x+14)
\end{aligned}$

$\begin{aligned}
H &= (3x-4)(2x+7)\\[0.5ex]
&\quad{}+(4-3x)(x-5)+6x-8\\[1ex]
&= (3x-4)(2x+7)\\[0.5ex]
&\quad{}+[-(3x-4)](x-5)\\[0.5ex]
&\quad{}+2\times3x-2\times4\\[1ex]
&= (3x-4)(2x+7)\\[0.5ex]
&\quad{}-(3x-4)(x-5)+2(3x-4)\\[1ex]
&= (3x-4)\times(2x+7)\\[0.5ex]
&\quad{}-(3x-4)\times(x-5)\\[0.5ex]
&\quad{}+(3x-4)\times2\\[1ex]
&= (3x-4)[(2x+7)-(x-5)+2]\\
&= (3x-4)(2x+7-x+5+2)\\
&= (3x-4)(x+14)
\end{aligned}$