Expressions algébriques/Exercices/Échauffement

Les exercices de cette page sont plus simples que les exercices des pages suivantes.

Exercice 1-1

Simplifier les expressions suivantes :

a)  $(x-y{)}^3-(x^3-y^3)$

b)  $\frac{x^2-y^2}{x^3-y^3}$

c)  $\frac{x^3-1}{x^3-3x^2+3x-1}$

d)  $\frac{x^3+27}{x^2+3x}$

e)  $\frac{(x^2+x+1)(x^2-x+1)}{x^6-1}$

f)  $\frac{x^6+2x^3{y}^3+y^6}{x^2+2xy+y^2}$

g)  $\frac{(x+1{)}^4-(x-1{)}^4}{8x^5+16x^3+8x}$

h)  $\frac{x^8-y^8}{x^6-x^4{y}^2+x^2{y}^4-y^6}$

i)  $\frac{x^3+1}{(x^2-1)(x^2-x+1)}$

a) Corrigé

$\begin{array}{cc}(x-y{)}^3-(x^3-y^3)& =(x-y{)}^3-(x-y)(x^2+xy+y^2)\\ & =(x-y)[(x-y{)}^2-(x^2+xy+y^2)]\\ & =(x-y)(x^2-2xy+y^2-x^2-xy-y^2)\\ & =-3xy(x-y)\end{array}$

 

b) Corrigé

$\begin{array}{cc}\frac{x^2-y^2}{x^3-y^3}& =\frac{(x-y)(x+y)}{(x-y)(x^2+xy+y^2)}\\ & =\frac{x+y}{x^2+xy+y^2}\end{array}$

 

c) Corrigé

$\begin{array}{cc}\frac{x^3-1}{x^3-3x^2+3x-1}& =\frac{(x-1)(x^2+x+1)}{(x-1{)}^3}\\ & =\frac{x^2+x+1}{(x-1{)}^2}\end{array}$

 

d) Corrigé

$\begin{array}{cc}\frac{x^3+27}{x^2+3x}& =\frac{(x+3)(x^2-3x+9)}{x(x+3)}\\ & =\frac{x^2-3x+9}{x}\end{array}$

 

e) Corrigé

$\begin{array}{cc}\frac{(x^2+x+1)(x^2-x+1)}{x^6-1}& =\frac{(x^2+x+1)(x^2-x+1)}{(x^3+1)(x^3-1)}\\ & =\frac{(x^2+x+1)(x^2-x+1)}{(x+1)(x^2-x+1)(x-1)(x^2+x+1)}\\ & =\frac{1}{(x+1)(x-1)}\\ & =\frac{1}{x^2-1}\end{array}$

 

f) Corrigé

$\begin{array}{cc}\frac{x^6+2x^3{y}^3+y^6}{x^2+2xy+y^2}& =\frac{(x^3+y^3{)}^2}{(x+y{)}^2}\\ & =\frac{(x+y{)}^2(x^2-xy+y^2{)}^2}{(x+y{)}^2}\\ & =(x^2-xy+y^2{)}^2\end{array}$

 

g) Corrigé

$\begin{array}{cc}\frac{(x+1{)}^4-(x-1{)}^4}{8x^5+16x^3+8x}& =\frac{[(x+1{)}^2-(x-1{)}^2][(x+1{)}^2+(x-1{)}^2]}{8x(x^4+2x^2+1)}\\ & =\frac{[(x+1)-(x-1)][(x+1)+(x-1)][(x^2+2x+1)+(x^2-2x+1)]}{8x(x^2+1{)}^2}\\ & =\frac{2\times 2x(2x^2+2)}{8x(x^2+1{)}^2}\\ & =\frac{8x(x^2+1)}{8x(x^2+1{)}^2}\\ & =\frac{1}{x^2+1}\end{array}$

Autre possibilité :

$\begin{array}{cc}\frac{(x+1{)}^4-(x-1{)}^4}{8x^5+16x^3+8x}& =\frac{(x^4+4x^3+6x^2+4x+1)-(x^4-4x^3+6x^2-4x+1)}{8x(x^4+2x^2+1)}\\ & =\frac{8x^3+8x}{8x(x^2+1{)}^2}\\ & =\frac{8x(x^2+1)}{8x(x^2+1{)}^2}\\ & =\frac{1}{x^2+1}\end{array}$

 

h) Corrigé

$\begin{array}{cc}\frac{x^8-y^8}{x^6-x^4{y}^2+x^2{y}^4-y^6}& =\frac{(x^4+y^4)(x^4-y^4)}{x^4(x^2-y^2)+y^4(x^2-y^2)}\\ & =\frac{(x^4+y^4)(x^2+y^2)(x^2-y^2)}{(x^4+y^4)(x^2-y^2)}\\ & =x^2+y^2\end{array}$

 

i) Corrigé

$\begin{array}{cc}\frac{x^3+1}{(x^2-1)(x^2-x+1)}& =\frac{(x+1)(x^2-x+1)}{(x+1)(x-1)(x^2-x+1)}\\ & =\frac{1}{x-1}\end{array}$

 

Exercice 1-2

Simplifier les expressions suivantes :

a)  $a(b+c-a{)}^2+b(c+a-b{)}^2+c(a+b-c{)}^2+(b+c-a)(c+a-b)(a+b-c)$

b)  $(2a^2+3ab-b^2{)}^2-4(a^2-b^2)(a^2+3ab+2b^2)$

c)  $(a^2+b^2+c^2+bc+ca+ab{)}^2-(a+b+c{)}^2(a^2+b^2+c^2)$

d)  $(abc+bcd+cda+dab{)}^2-(bc-ad)(ca-bd)(ab-cd)$

a) Corrigé

On trouve :

$4abc$

 

b) Corrigé

On trouve :

$b^2(a+3b{)}^2$

 

c) Corrigé

On peut simplifier le calcul en posant :

$A=a^2+b^2+c^2$

$B=bc+ca+ab$

On a alors :

$\begin{array}{cc}(a^2+b^2+c^2+bc+ca+ab{)}^2-(a+b+c{)}^2(a^2+b^2+c^2)& =(a^2+b^2+c^2+bc+ca+ab{)}^2-(a^2+b^2+c^2+2ab+2ac+2bc)(a^2+b^2+c^2)\\ & =(A+B{)}^2-(A+2B)(A)\\ & =A^2+2AB+B^2-A^2-2AB\\ & =B^2\\ & =(bc+ca+ab{)}^2\end{array}$

 

d) Corrigé

On a :

$(abc+bcd+cda+dab{)}^2=a^2{b}^2{c}^2+b^2{c}^2{d}^2+c^2{d}^2{a}^2+d^2{a}^2{b}^2+abcd(2ab+2ac+2ad+2bc+2bd+2cd)$

$(bc-ad)(ca-bd)(ab-cd)=a^2{b}^2{c}^2+b^2{c}^2{d}^2+c^2{d}^2{a}^2+d^2{a}^2{b}^2-abcd(a^2+b^2+c^2+d^2)$

Par différence membre à membre, on obtient :

$\begin{array}{cc}(abc+bcd+cda+dab{)}^2-(bc-ad)(ca-bd)(ab-cd)& =abcd(2ab+2ac+2ad+2bc+2bd+2cd)+abcd(a^2+b^2+c^2+d^2)\\ & =abcd(a^2+b^2+c^2+d^2+2ab+2ac+2ad+2bc+2bd+2cd)\\ & =abcd(a+b+c+d{)}^2\end{array}$

 

Exercice 1-3

Factoriser les expressions suivantes :

a)  $a^3-3a^{2}b+3a{b}^2-b^3$

b)  $a^3-a^{2}b-a{b}^2+b^3$

a) Corrigé

$a^3-3a^{2}b+3a{b}^2-b^3=(a-b{)}^3$

 

b) Corrigé

$\begin{array}{cc}{a}^3-a^{2}b-a{b}^2+b^3& =a^3+b^3-a^{2}b-a{b}^2\\ & =(a+b)(a^2-ab+b^2)-ab(a+b)\\ & =(a+b)(a^2-ab+b^2-ab)\\ & =(a+b)(a^2-2ab+b^2)\\ & =(a+b)(a-b{)}^2\end{array}$

ou

$\begin{array}{cc}{a}^3-a^{2}b-a{b}^2+b^3& =a^2(a-b)-b^2(a-b)\\ & =(a^2-b^2)(a-b)\\ & =(a+b)(a-b)(a-b)\\ & =(a+b)(a-b{)}^2\end{array}$

 

Exercice 1-4

Simplifier les expressions suivantes :

a)  $\sqrt{a^2+6a+9}+\sqrt{a^2-10a+25}$

b)  $\sqrt{2}\sqrt{2+\sqrt{2}}\sqrt{2-\sqrt{2}}$

c)  $(\sqrt{2}+\sqrt{3}+\sqrt{5})(\sqrt{2}-\sqrt{3}+\sqrt{5})(\sqrt{5}-\sqrt{2})$

d)  $\frac{3\sqrt{8}-2\sqrt{12}+\sqrt{20}}{3\sqrt{18}-2\sqrt{27}+\sqrt{45}}$

e)  $\frac{3}{\sqrt{5}-\sqrt{2}}+\frac{4}{\sqrt{6}+\sqrt{2}}$

a) Corrigé

$\begin{array}{cc}\sqrt{a^2+6a+9}+\sqrt{a^2-10a+25}& =\sqrt{(a+3{)}^2}+\sqrt{(a-5{)}^2}\\ & =a+3+a-5\\ & =2a-2\\ & =2(a-1)\end{array}$

 

b) Corrigé

$\begin{array}{cc}\sqrt{2}\sqrt{2+\sqrt{2}}\sqrt{2-\sqrt{2}}& =\sqrt{2}\sqrt{(2+\sqrt{2})(2-\sqrt{2})}\\ & =\sqrt{2}\sqrt{2^2-{\left(\sqrt{2}\right)}^2}\\ & =\sqrt{2}\sqrt{4-2}\\ & =\sqrt{2}\sqrt{2}\\ & =2\end{array}$

 

c) Corrigé

$\begin{array}{cc}(\sqrt{2}+\sqrt{3}+\sqrt{5})(\sqrt{2}-\sqrt{3}+\sqrt{5})(\sqrt{5}-\sqrt{2})& =(2-\sqrt{6}+\sqrt{10}+\sqrt{6}-3+\sqrt{15}+\sqrt{10}-\sqrt{15}+5)(\sqrt{5}-\sqrt{2})\\ & =(4+2\sqrt{10})(\sqrt{5}-\sqrt{2})\\ & =4\sqrt{5}-4\sqrt{2}+2\sqrt{50}-2\sqrt{20}\\ & =4\sqrt{5}-4\sqrt{2}+10\sqrt{2}-4\sqrt{5}\\ & =6\sqrt{2}\end{array}$

 

d) Corrigé

$\begin{array}{cc}\frac{3\sqrt{8}-2\sqrt{12}+\sqrt{20}}{3\sqrt{18}-2\sqrt{27}+\sqrt{45}}& =\frac{6\sqrt{2}-4\sqrt{3}+2\sqrt{5}}{9\sqrt{2}-6\sqrt{3}+3\sqrt{5}}\\ & =\frac{2(3\sqrt{2}-2\sqrt{3}+\sqrt{5})}{3(3\sqrt{2}-2\sqrt{3}+\sqrt{5})}\\ & =\frac{2}{3}\end{array}$

 

e) Corrigé

$\begin{array}{cc}\frac{3}{\sqrt{5}-\sqrt{2}}+\frac{4}{\sqrt{6}+\sqrt{2}}& =\frac{3(\sqrt{5}+\sqrt{2})}{(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})}+\frac{4(\sqrt{6}-\sqrt{2})}{(\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})}\\ & =\frac{3(\sqrt{5}+\sqrt{2})}{5-2}+\frac{4(\sqrt{6}-\sqrt{2})}{6-2}\\ & =\frac{3(\sqrt{5}+\sqrt{2})}{3}+\frac{4(\sqrt{6}-\sqrt{2})}{4}\\ & =\sqrt{5}+\sqrt{2}+\sqrt{6}-\sqrt{2}\\ & =\sqrt{5}+\sqrt{6}\end{array}$