Calcul littéral, développements et factorisations

A) \(\;a(a+5)\)

\[ a(a+5)=a\cdot a+a\cdot 5=a^2+5a \]

B) \(\;(x^2+6)x\)

\[ (x^2+6)x=x^2\cdot x+6\cdot x=x^3+6x \]

C) \(\;a(2a^2-3a)\)

\[ a(2a^2-3a)=a\cdot 2a^2-a\cdot 3a=2a^3-3a^2 \]

D) \(\;a(3a^2-2)\)

\[ a(3a^2-2)=a\cdot 3a^2-a\cdot 2=3a^3-2a \]

E) \(\;b(4b+7)\)

\[ b(4b+7)=b\cdot 4b+b\cdot 7=4b^2+7b \]

F) \(\;(9c^2-5c)c\)

\[ (9c^2-5c)c=9c^2\cdot c-5c\cdot c=9c^3-5c^2 \]

A) \(\;x^2(3x-2x^2)\)

\[ x^2(3x-2x^2)=x^2\cdot 3x-x^2\cdot 2x^2=3x^3-2x^4 \]

B) \(\;(4x^2-7)\,2x^2\)

\[ (4x^2-7)\,2x^2=4x^2\cdot 2x^2-7\cdot 2x^2=8x^4-14x^2 \]

C) \(\;5a^3(2a^3-3a)\)

\[ 5a^3(2a^3-3a)=5a^3\cdot 2a^3-5a^3\cdot 3a=10a^6-15a^4 \]

D) \(\;(6a-4a^2)a^2\)

\[ (6a-4a^2)a^2=6a\cdot a^2-4a^2\cdot a^2=6a^3-4a^4 \]

E) \(\;3a^2(5x+8)\)

\[ 3a^2(5x+8)=3a^2\cdot 5x+3a^2\cdot 8=15a^2x+24a^2 \]

F) \(\;2x(7x^2-5x)\)

\[ 2x(7x^2-5x)=2x\cdot 7x^2-2x\cdot 5x=14x^3-10x^2 \]

A) \(\;3xy(x^2y-2x)\)

\[ 3xy(x^2y-2x)=3xy\cdot x^2y-3xy\cdot 2x \] \[ =3x^3y^2-6x^2y \]

B) \(\;(5ab-3ab^2)\,2a^2b\)

\[ (5ab-3ab^2)\,2a^2b=5ab\cdot 2a^2b-3ab^2\cdot 2a^2b \] \[ =10a^3b^2-6a^3b^3 \]

C) \(\;4y^2(y^3+3x^2y-2)\)

\[ 4y^2(y^3+3x^2y-2)=4y^2\cdot y^3+4y^2\cdot 3x^2y-4y^2\cdot 2 \] \[ =4y^5+12x^2y^3-8y^2 \]

D) \(\;(2a^3-3a^2b+4)\,3ab\)

\[ (2a^3-3a^2b+4)\,3ab=2a^3\cdot 3ab-3a^2b\cdot 3ab+4\cdot 3ab \] \[ =6a^4b-9a^3b^2+12ab \]

E) \(\;x^3(4xy-3x)\)

\[ x^3(4xy-3x)=x^3\cdot 4xy-x^3\cdot 3x=4x^4y-3x^4 \]

F) \(\;(3a^2b-5b)\,ab\)

\[ (3a^2b-5b)\,ab=3a^2b\cdot ab-5b\cdot ab=3a^3b^2-5ab^2 \]

A) \(\;(4t-3)(6t-5)\)

\[ (4t-3)(6t-5)=4t\cdot 6t+4t\cdot(-5)+(-3)\cdot 6t+(-3)\cdot(-5) \] \[ =24t^2-20t-18t+15=24t^2-38t+15 \]

B) \(\;(3s+2)(5s-7)\)

\[ (3s+2)(5s-7)=3s\cdot 5s+3s\cdot(-7)+2\cdot 5s+2\cdot(-7) \] \[ =15s^2-21s+10s-14=15s^2-11s-14 \]

C) \(\;(4x-1)(3x+2)\)

\[ (4x-1)(3x+2)=4x\cdot 3x+4x\cdot 2+(-1)\cdot 3x+(-1)\cdot 2 \] \[ =12x^2+8x-3x-2=12x^2+5x-2 \]

D) \(\;(5y-4)(3y-2)\)

\[ (5y-4)(3y-2)=5y\cdot 3y+5y\cdot(-2)+(-4)\cdot 3y+(-4)\cdot(-2) \] \[ =15y^2-10y-12y+8=15y^2-22y+8 \]

E) \(\;(2x+3y)(x-2y)\)

\[ (2x+3y)(x-2y)=2x\cdot x+2x\cdot(-2y)+3y\cdot x+3y\cdot(-2y) \] \[ =2x^2-4xy+3xy-6y^2=2x^2-xy-6y^2 \]

F) \(\;(5x-2y)(2x-3y)\)

\[ (5x-2y)(2x-3y)=5x\cdot 2x+5x\cdot(-3y)+(-2y)\cdot 2x+(-2y)\cdot(-3y) \] \[ =10x^2-15xy-4xy+6y^2=10x^2-19xy+6y^2 \]

A) \(\;(2x^2-3)(5x^2+4)\)

\[ (2x^2-3)(5x^2+4)=2x^2\cdot 5x^2+2x^2\cdot 4+(-3)\cdot 5x^2+(-3)\cdot 4 \] \[ =10x^4+8x^2-15x^2-12=10x^4-7x^2-12 \]

B) \(\;(a^2b-2a)(3a^2b+a)\)

\[ (a^2b-2a)(3a^2b+a)=a^2b\cdot 3a^2b+a^2b\cdot a+(-2a)\cdot 3a^2b+(-2a)\cdot a \] \[ =3a^4b^2+a^3b-6a^3b-2a^2=3a^4b^2-5a^3b-2a^2 \]

C) \(\;(4ab-3b)(2ab+5b)\)

\[ (4ab-3b)(2ab+5b)=4ab\cdot 2ab+4ab\cdot 5b+(-3b)\cdot 2ab+(-3b)\cdot 5b \] \[ =8a^2b^2+20ab^2-6ab^2-15b^2=8a^2b^2+14ab^2-15b^2 \]

D) \(\;(2y^2-7x)(4x+3y^2)\)

\[ (2y^2-7x)(4x+3y^2)=2y^2\cdot 4x+2y^2\cdot 3y^2+(-7x)\cdot 4x+(-7x)\cdot 3y^2 \] \[ =8xy^2+6y^4-28x^2-21xy^2=6y^4-13xy^2-28x^2 \]

E) \(\;(3x^2-2x)(-5x+2x^2)\)

\[ (3x^2-2x)(-5x+2x^2)=3x^2\cdot(-5x)+3x^2\cdot 2x^2+(-2x)\cdot(-5x)+(-2x)\cdot 2x^2 \] \[ =-15x^3+6x^4+10x^2-4x^3=6x^4-19x^3+10x^2 \]

F) \(\;(-3x^2-2y)(-2x-3y^2)\)

\[ (-3x^2-2y)(-2x-3y^2)=(-3x^2)\cdot(-2x)+(-3x^2)\cdot(-3y^2)+(-2y)\cdot(-2x)+(-2y)\cdot(-3y^2) \] \[ =6x^3+9x^2y^2+4xy+6y^3 \]

A) \(\;(a^3-4b)(-5a+2b^2)\)

\[ (a^3-4b)(-5a+2b^2)=a^3\cdot(-5a)+a^3\cdot 2b^2+(-4b)\cdot(-5a)+(-4b)\cdot 2b^2 \] \[ =-5a^4+2a^3b^2+20ab-8b^3 \]

B) \(\;(4abc-3ab)(10ab-8abc)\)

\[ (4abc-3ab)(10ab-8abc)=4abc\cdot 10ab+4abc\cdot(-8abc)+(-3ab)\cdot 10ab+(-3ab)\cdot(-8abc) \] \[ =40a^2b^2c-32a^2b^2c^2-30a^2b^2+24a^2b^2c \] \[ =a^2b^2(-32c^2+64c-30) \]

C) \(\;(2ab^2+5a^2b)(-3a^2b+4ab^2)\)

\[ (2ab^2+5a^2b)(-3a^2b+4ab^2)=2ab^2\cdot(-3a^2b)+2ab^2\cdot 4ab^2+5a^2b\cdot(-3a^2b)+5a^2b\cdot 4ab^2 \] \[ =-6a^3b^3+8a^2b^4-15a^4b^2+20a^3b^3 \] \[ =8a^2b^4+14a^3b^3-15a^4b^2=a^2b^2(8b^2+14ab-15a^2) \]

D) \(\;(3a^3b-5ab^3)(-2a^3b+ab^3)\)

\[ (3a^3b-5ab^3)(-2a^3b+ab^3)=3a^3b\cdot(-2a^3b)+3a^3b\cdot ab^3+(-5ab^3)\cdot(-2a^3b)+(-5ab^3)\cdot ab^3 \] \[ =-6a^6b^2+3a^4b^4+10a^4b^4-5a^2b^6 \] \[ =-6a^6b^2+13a^4b^4-5a^2b^6 \]

A) \(\;(6x-5y)^2\) — identité \((a-b)^2=a^2-2ab+b^2\)

\[ (6x-5y)^2=(6x)^2-2\cdot(6x)\cdot(5y)+(5y)^2 \] \[ =36x^2-60xy+25y^2 \]

B) \(\;(3a-4b)^2\)

\[ (3a-4b)^2=(3a)^2-2\cdot(3a)\cdot(4b)+(4b)^2 \] \[ =9a^2-24ab+16b^2 \]

C) \(\;(2a+5b)^2\) — identité \((a+b)^2=a^2+2ab+b^2\)

\[ (2a+5b)^2=(2a)^2+2\cdot(2a)\cdot(5b)+(5b)^2 \] \[ =4a^2+20ab+25b^2 \]

D) \(\;(5x-9y)^2\)

\[ (5x-9y)^2=(5x)^2-2\cdot(5x)\cdot(9y)+(9y)^2 \] \[ =25x^2-90xy+81y^2 \]

E) \(\;(3b-8c)^2\)

\[ (3b-8c)^2=(3b)^2-2\cdot(3b)\cdot(8c)+(8c)^2 \] \[ =9b^2-48bc+64c^2 \]

F) \(\;(4x-3y)(4x+3y)\) — identité \((a-b)(a+b)=a^2-b^2\)

\[ (4x-3y)(4x+3y)=(4x)^2-(3y)^2=16x^2-9y^2 \]

A) \(\;(4a-3b)^2\)

\[ (4a-3b)^2=(4a)^2-2\cdot(4a)\cdot(3b)+(3b)^2 \] \[ =16a^2-24ab+9b^2 \]

B) \(\;(3-2b)^2\) — attention : \(3-2b=3+(-2b)\)

\[ (3-2b)^2=3^2-2\cdot 3\cdot(2b)+(2b)^2 \] \[ =9-12b+4b^2 \]

C) \(\;(5a+2b)^2\)

\[ (5a+2b)^2=(5a)^2+2\cdot(5a)\cdot(2b)+(2b)^2 \] \[ =25a^2+20ab+4b^2 \]

D) \(\;(4x-z)(4x+z)\)

\[ (4x-z)(4x+z)=(4x)^2-z^2=16x^2-z^2 \]

E) \(\;(3a-8)^2\)

\[ (3a-8)^2=(3a)^2-2\cdot(3a)\cdot 8+8^2 \] \[ =9a^2-48a+64 \]

F) \(\;(8a-5b)^2\)

\[ (8a-5b)^2=(8a)^2-2\cdot(8a)\cdot(5b)+(5b)^2 \] \[ =64a^2-80ab+25b^2 \]

A)

On repère \((x+a)(x-a)\) : forme \((A+B)(A-B)=A^2-B^2\).

\[ (x+a)(x-a)=x^2-a^2 \] \[ A=(x+a)(x-a)(x^2-a^2)=(x^2-a^2)(x^2-a^2)=(x^2-a^2)^2 \] \[ (x^2-a^2)^2=x^4-2a^2x^2+a^4 \]

B)

\[ (3a-2)(3a+2)=(3a)^2-2^2=9a^2-4 \] \[ (9a^2-4)(9a^2+4)=(9a^2)^2-4^2=81a^4-16 \] \[ \boxed{B=81a^4-16} \]

C)

\[ (x-2)(x+2)=x^2-4 \] \[ (x^2-4)(x^2+4)=x^4-16 \] \[ \boxed{C=x^4-16} \]

D)

\[ (x+3)(x-3)=x^2-9 \] \[ (x^2-9)(x^2+9)=x^4-81 \] \[ (x^4-81)(x^4+81)=(x^4)^2-81^2=x^8-6561 \] \[ \boxed{D=x^8-6561} \]

E)

\[ (x^2-4)(x^2+4)=x^4-16 \] \[ (x^4-16)(x^4-5)=x^8-5x^4-16x^4+80=x^8-21x^4+80 \] \[ \boxed{E=x^8-21x^4+80} \]

F)

\[ (3a^2+1)(3a^2-1)=(3a^2)^2-1^2=9a^4-1 \] \[ (9a^4+2)(9a^4-1)=(9a^4)^2+(2-1)\cdot 9a^4-2 \] \[ =81a^8+9a^4-2 \] \[ \boxed{F=81a^8+9a^4-2} \]

A) forme \((U+V)(U-V)=U^2-V^2\)

\[ U=x-2,\quad V=x^2 \] \[ A=U^2-V^2=(x-2)^2-(x^2)^2 \] \[ (x-2)^2=x^2-4x+4 \] \[ A=(x^2-4x+4)-x^4=-x^4+x^2-4x+4 \]

B) même forme \((U+V)(U-V)=U^2-V^2\)

\[ U=x,\quad V=3+x^2 \] \[ B=x^2-(3+x^2)^2 \] \[ (3+x^2)^2=x^4+6x^2+9 \] \[ B=x^2-(x^4+6x^2+9)=-x^4-5x^2-9 \]

C) on regroupe pour faire apparaître \((x+(y-2))(x-(y-2))\)

\[ (x+y-2)(x-y+2)=(x+(y-2))(x-(y-2))=x^2-(y-2)^2 \] \[ (y-2)^2=y^2-4y+4 \] \[ C=x^2-(y^2-4y+4)=x^2-y^2+4y-4 \]

D) on reconnaît deux carrés : \((a-b)^2\) et \((a+b)^2\)

\[ a^2-2ab+b^2=(a-b)^2,\quad a^2+2ab+b^2=(a+b)^2 \] \[ D=(a-b)^2(a+b)^2=((a-b)(a+b))^2=(a^2-b^2)^2 \] \[ (a^2-b^2)^2=a^4-2a^2b^2+b^4 \]

Étape 1 — Substitution intelligente

\[ p=\frac{u}{v},\quad q=\frac{v}{w},\quad r=\frac{w}{u} \]

Alors :

\[ pqr=\frac{u}{v}\cdot\frac{v}{w}\cdot\frac{w}{u}=1 \]

Et :

\[ a=p+\frac{1}{p},\quad b=q+\frac{1}{q},\quad c=r+\frac{1}{r} \]

Étape 2 — Utiliser l’identité (cas \(pqr=1\))

\[ \left(p+\frac{1}{p}\right)^2+\left(q+\frac{1}{q}\right)^2+\left(r+\frac{1}{r}\right)^2 -\left(p+\frac{1}{p}\right)\left(q+\frac{1}{q}\right)\left(r+\frac{1}{r}\right)=4 \]

Conclusion

\[ \boxed{a^2+b^2+c^2-abc=4} \]

1) Calcul de \(XY\)

\[ (X+Y)^2=X^2+2XY+Y^2 \] \[ 5^2=29+2XY \Rightarrow 25=29+2XY \] \[ 2XY=25-29=-4 \Rightarrow \boxed{XY=-2} \]

2) Calcul de \(\frac{1}{X}+\frac{1}{Y}\)

\[ \frac{1}{X}+\frac{1}{Y}=\frac{X+Y}{XY} \] \[ \frac{1}{X}+\frac{1}{Y}=\frac{5}{-2}=\boxed{-\frac{5}{2}} \]

3) Calcul de \(X^3+Y^3\)

\[ X^3+Y^3=(X+Y)^3-3XY(X+Y) \] \[ X^3+Y^3=5^3-3(-2)\cdot 5=125+30=\boxed{155} \]

4) Calcul de \(X^4+Y^4\)

\[ X^4+Y^4=(X^2+Y^2)^2-2X^2Y^2 \] \[ X^2Y^2=(XY)^2=(-2)^2=4 \] \[ X^4+Y^4=29^2-2\cdot 4=841-8=\boxed{833} \]