Identités remarquables/Exercices/Factorisation

**Factorisation**
Leçon : Identités remarquables

Page d'exercices n^o 4
Chapitre du cours :	Factorisation
Exercices de niveau 10.
Exo préc. :	Développement
Exo suiv. :	Quotient

En raison de limitations techniques, la typographie souhaitable du titre, « Exercice : Factorisation
Identités remarquables/Exercices/Factorisation », n'a pu être restituée correctement ci-dessus.

Exercice 4-1

Factoriser en utilisant les identités remarquables.

a) $x^{2} + 14 x + 49$

b) $9 x^{2} - 6 x + 1$

c) $2 x^{2} + x \sqrt{24} + 3$

d) $\frac{1}{4} - 3 x + 9 x^{2}$

e) $x^{2} - 25$

f) $9 x^{2} - 16$

g) $3 x^{2} - 1$

h) $(x - 3)^{2} - 16$

i) $(x - 3)^{2} - 2$

j) $4 (x - 3)^{2} - 25 (1 - 3 x)^{2}$

Solution

a) $x^{2} + 14 x + 49 = x^{2} + 2 \times 7 x + 7^{2} = (x + 7)^{2}$

b) $9 x^{2} - 6 x + 1 = (3 x)^{2} - 2 \times 3 x + 1^{2} = (3 x + 1)^{2}$

c) $2 x^{2} + x \sqrt{24} + 3 = 2 x^{2} + 2 x \sqrt{6} + 3 = (x \sqrt{2})^{2} + 2 \times x \sqrt{2} \times \sqrt{3} + (\sqrt{3})^{2} = (x \sqrt{2} + \sqrt{3})^{2}$

d) $\frac{1}{4} - 3 x + 9 x^{2} = {(\frac{1}{2})}^{2} - 2 \times \frac{1}{2} 3 x + (3 x)^{2} = {(\frac{1}{2} - 3 x)}^{2}$

e) $x^{2} - 25 = x^{2} - 5^{2} = (x + 5) (x - 5)$

f) $9 x^{2} - 16 = (3 x)^{2} - 4^{2} = (3 x + 4) (3 x - 4)$

g) $3 x^{2} - 1 = (x \sqrt{3})^{2} - 1^{2} = (x \sqrt{3} + 1) (x \sqrt{3} - 1)$

h) $(x - 3)^{2} - 16 = (x - 3)^{2} - 4^{2} = (x - 3 + 4) (x - 3 - 4) = (x + 1) (x - 7)$

i) $(x - 3)^{2} - 2 = (x - 3)^{2} - (\sqrt{2})^{2} = (x - 3 + \sqrt{2}) (x - 3 - \sqrt{2})$

j) $4 (x - 3)^{2} - 25 (1 - 3 x)^{2} = [2 (x - 3)]^{2} - [5 (1 - 3 x)]^{2} = [2 (x - 3) + 5 (1 - 3 x)] [2 (x - 3) - 5 (1 - 3 x)] = (2 x - 6 + 5 - 15 x) (2 x - 6 - 5 + 15 x) = (- 13 x - 1) (17 x - 11)$

Exercice 4-2

Factoriser les expressions suivantes :

a) $(3 x + 1) (- 4 x + 5) - 2 (3 x + 1)$

b) $(x - 5) (x + 3) - (2 x + 7) (x + 3)$

c) $(x + 2) (5 x - 1) - (5 x + 2) (2 x + 4)$

d) $(2 x + 3)^{2} - (4 x + 6) (- 4 x + 2)$

Solution

a) $(3 x + 1) (- 4 x + 5) - 2 (3 x + 1) = (3 x + 1) [(- 4 x + 5) - 2] = (3 x + 1) (- 4 x + 3)$

b) $(x - 5) (x + 3) - (2 x + 7) (x + 3) = (x + 3) [(x - 5) - (2 x + 7)] = (x + 3) (- x - 12)$

c) $(x + 2) (5 x - 1) - (5 x + 2) (2 x + 4) = (x + 2) (5 x - 1) - 2 (5 x + 2) (x + 2) = (x + 2) [(5 x - 1) - 2 (5 x + 2)] = (x + 2) (5 x - 1 - 10 x - 4) = (x + 2) (- 5 x - 5) = - 5 (x + 2) (x + 1)$

d) $(2 x + 3)^{2} - (4 x + 6) (- 4 x + 2) = (2 x + 3)^{2} - 2 (2 x + 3) (- 4 x + 2) = (2 x + 3) [(2 x + 3) - 2 (- 4 x + 2)] = (2 x + 3) (2 x + 3 + 8 x - 4) = (2 x + 3) (10 x - 1)$

Exercice 4-3

Factoriser les expressions suivantes :

a) $4 x^{2} - 1 - (2 x - 1) (3 x + 5)$

b) $4 x^{2} + 4 x + 1 - (2 x + 1) (3 x + 5)$

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

d) $3 x (x - 1) - 6 (x^{2} - 2 x + 1) + 9 x (x^{2} - 1)$

e) $2 x y (y^{2} - 9) - 4 x (3 - y) + 8 x (y^{2} - 6 y + 9)$

Solution

$\begin{matrix} a) 4 x^{2} - 1 - (2 x - 1) (3 x + 5) & = (2 x)^{2} - 1 - (2 x - 1) (3 x + 5) \\ = (2 x + 1) (2 x - 1) - (2 x - 1) (3 x + 5) \\ = (2 x - 1) [(2 x + 1) - (3 x + 5)] \\ = (2 x - 1) (- x - 4) \\ = - (2 x - 1) (x + 4) \end{matrix}$

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

$\begin{matrix} c) a b (a + b) + b c (a^{2} + 2 a b + b^{2}) + b^{2} (a^{2} - b^{2}) & = a b (a + b) + b c (a + b)^{2} + b^{2} (a + b) (a - b) \\ = b (a + b) [a + c (a + b) + b (a - b)] \\ = b (a + b) (a + a c + b c + a b - b^{2}) \end{matrix}$

$\begin{matrix} d) 3 x (x - 1) - 6 (x^{2} - 2 x + 1) + 9 x (x^{2} - 1) & = 3 x (x - 1) - 6 (x - 1)^{2} + 9 x (x + 1) (x - 1) \\ = 3 (x - 1) [x - 2 (x - 1) + 3 x (x + 1)] \\ = 3 (x - 1) (3 x^{2} + 2 x + 2) \end{matrix}$

$\begin{matrix} e) 2 x y (y^{2} - 9) - 4 x (3 - y) + 8 x (y^{2} - 6 y + 9) & = 2 x y (y + 3) (y - 3) + 4 x (y - 3) + 8 x (y - 3)^{2} \\ = 2 x (y - 3) [y (y + 3) + 2 + 4 (y - 3)] \\ = 2 x (y - 3) [y^{2} + 3 y + 2 + 4 y - 12] \\ = 2 x (y - 3) (y^{2} + 7 y - 10) \end{matrix}$

Exercice 4-4

Factoriser les expressions suivantes :

a) $(x + 3 y)^{2} - (x - 5 y)^{2}$

b) $(x^{2} + 2 x y + y^{2}) - (m^{2} - 2 m n + n^{2})$

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

d) $a^{2} - b^{2} - c^{2} - 2 b c$

Solution

$\begin{matrix} a) (x + 3 y)^{2} - (x - 5 y)^{2} & = (x + 3 y + x - 5 y) (x + 3 y - x + 5 y) \\ = (2 x - 2 y) (8 y) \\ = 16 y (x - y) \end{matrix}$

$\begin{matrix} b) (x^{2} + 2 x y + y^{2}) - (m^{2} - 2 m n + n^{2}) & = (x + y)^{2} - (m - n)^{2} \\ = (x + y + m - n) (x + y - m + n) \end{matrix}$

$\begin{matrix} c) (5 + 2 \sqrt{5 a} + a) - (3 - 2 \sqrt{3 b} + b) & = ((\sqrt{5})^{2} + 2 \sqrt{5} \sqrt{a} + (\sqrt{a})^{2}) - ((\sqrt{3})^{2} - 2 \sqrt{3} \sqrt{b} + (\sqrt{b})^{2}) \\ = (\sqrt{5} + \sqrt{a})^{2} - (\sqrt{3} - \sqrt{b})^{2} \\ = (\sqrt{5} + \sqrt{a} + \sqrt{3} - \sqrt{b}) (\sqrt{5} + \sqrt{a} - \sqrt{3} + \sqrt{b}) \end{matrix}$

$\begin{matrix} d) a^{2} - b^{2} - c^{2} - 2 b c & = a^{2} - (b^{2} + c^{2} + 2 b c) \\ = a^{2} - {(b + c)}^{2} \\ = (a + b + c) (a - b - c) \end{matrix}$

Exercice 4-5

Factoriser les expressions suivantes :

a) $a b c^{2} - a b d^{2} + c e - d e$

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

c) $x^{2} + 2 + \frac{1}{x^{2}}$

d) $a x + a y + b x + b y + (a - b)^{2} + 4 a b$

Solution

$\begin{matrix} a) a b c^{2} - a b d^{2} + c e - d e & = a b (c^{2} - d^{2}) + e (c - d) \\ = a b (c + d) (c - d) + e (c - d) \\ = (c - d) [a b (c + d) + e] \\ = (c - d) (a b c + a b d + e) \end{matrix}$

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

$\begin{matrix} c) x^{2} + 2 + \frac{1}{x^{2}} & = x^{2} + 2 x \frac{1}{x} + {(\frac{1}{x})}^{2} \\ = {(x + \frac{1}{x})}^{2} \end{matrix}$

$\begin{matrix} d) a x + a y + b x + b y + (a - b)^{2} + 4 a b & = a (x + y) + b (x + y) + a^{2} - 2 a b + b^{2} + 4 a b \\ = (a + b) (x + y) + a^{2} + 2 a b + b^{2} \\ = (a + b) (x + y) + (a + b)^{2} \\ = (a + b) (x + y + a + b) \end{matrix}$

Exercice 4-6

Factoriser les expressions suivantes :

a) $(x + 2 \sqrt{3} - 5)^{2} - (x - 3 \sqrt{3} + 2)^{2}$

b) $x^{2} + 2 x (x - 1) + (x - 1)^{2}$

c) $m^{2} + \frac{1}{m^{2}} + 1$

d) $4 x^{2} + 4 x (x + 3) + (x + 3)^{2} + 6 (x + 1)$

Solution

$\begin{matrix} a) (x + 2 \sqrt{3} - 5)^{2} - (x - 3 \sqrt{3} + 2)^{2} & = (x + 2 \sqrt{3} - 5 + x - 3 \sqrt{3} + 2) (x + 2 \sqrt{3} - 5 - x + 3 \sqrt{3} - 2) \\ = (2 x - \sqrt{3} - 3) (5 \sqrt{3} - 7) \end{matrix}$

$\begin{matrix} b) x^{2} + 2 x (x - 1) + (x - 1)^{2} & = [x + (x - 1)]^{2} \\ = (2 x - 1)^{2} \end{matrix}$

$\begin{matrix} c) m^{2} + \frac{1}{m^{2}} + 1 & = m^{2} + 2 + \frac{1}{m^{2}} - 1 \\ = m^{2} + 2 m . \frac{1}{m} + {(\frac{1}{m})}^{2} - 1^{2} \\ = (m + \frac{1}{m})^{2} - 1^{2} \\ = (m + \frac{1}{m} + 1) (m + \frac{1}{m} - 1) \end{matrix}$

$\begin{matrix} d) 4 x^{2} + 4 x (x + 3) + (x + 3)^{2} + 6 (x + 1) & = (2 x)^{2} + 2 \times 2 x (x + 3) + (x + 3)^{2} + 6 (x + 1) \\ = (2 x + x + 3)^{2} + 6 (x + 1) \\ = (3 x + 3)^{2} + 6 (x + 1) \\ = 9 (x + 1)^{2} + 6 (x + 1) \\ = (x + 1) [9 (x + 1) + 6] \\ = (x + 1) (9 x + 15) \\ = 3 (x + 1) (3 x + 5) \end{matrix}$

Exercice 4-7

Factoriser les expressions suivantes :

a) $x^{2} - 8 x + 15$

b) $x^{2} + 2 x - 35$

c) $x^{2} + x - 6$

d) $x^{2} - 3 x - 40$

e) $x^{2} + 6 x + 7$

Solution

$\begin{matrix} a) x^{2} - 8 x + 15 & = x^{2} - 8 x + 16 - 1 \\ = x^{2} - 8 x + 4^{2} - 1^{2} \\ = (x - 4)^{2} - 1^{2} \\ = (x - 4 + 1) (x - 4 - 1) \\ = (x - 3) (x - 5) \end{matrix}$

$\begin{matrix} b) x^{2} + 2 x - 35 & = x^{2} + 2 x + 1^{2} - 6^{2} \\ = (x + 1)^{2} - 6^{2} \\ = (x + 1 + 6) (x + 1 - 6) \\ = (x + 7) (x - 5) \end{matrix}$

$\begin{matrix} c) x^{2} + x - 6 & = {(x + \frac{1}{2})}^{2} - \frac{1}{4} - 6 \\ = {(x + \frac{1}{2})}^{2} - \frac{25}{4} \\ = {(x + \frac{1}{2})}^{2} - {(\frac{5}{2})}^{2} \\ = (x + \frac{1}{2} + \frac{5}{2}) (x + \frac{1}{2} - \frac{5}{2}) \\ = (x + 3) (x - 2) \end{matrix}$

$\begin{matrix} d) x^{2} - 3 x - 40 & = {(x - \frac{3}{2})}^{2} - \frac{9}{4} - 40 \\ = {(x - \frac{3}{2})}^{2} - {(\frac{13}{2})}^{2} \\ = (x - \frac{3}{2} + \frac{13}{2}) (x - \frac{3}{2} - \frac{13}{2}) \\ = (x + 5) (x - 8) \end{matrix}$

$\begin{matrix} e) x^{2} + 6 x + 7 & = (x + 3)^{2} - 2 \\ = (x + 3)^{2} - (\sqrt{2})^{2} \\ = (x + 3 + \sqrt{2}) (x + 3 - \sqrt{2}) \end{matrix}$

Exercice 4-8

Factoriser les expressions suivantes :

a) $3 (x^{2} + 2 x - 35) - (x + 2) (x - 5)$

b) $(3 x - 1)^{2} - (2 x + 3)^{2} - (x - 4)^{2}$

Solution

$\begin{matrix} a) 3 (x^{2} + 2 x - 35) - (x + 2) (x - 5) & = 3 [(x + 1)^{2} - 6^{2}] - (x + 2) (x - 5) \\ = 3 [(x + 1 + 6) (x + 1 - 6)] - (x + 2) (x - 5) \\ = 3 (x + 7) (x - 5) - (x + 2) (x - 5) \\ = (x - 5) [3 (x + 7) - (x + 2)] \\ = (x - 5) (2 x + 19) \end{matrix}$

$\begin{matrix} b) (3 x - 1)^{2} - (2 x + 3)^{2} - (x - 4)^{2} & = [(3 x - 1) + (2 x + 3)] [(3 x - 1) - (2 x + 3)] - (x - 4)^{2} \\ = (3 x - 1 + 2 x + 3) (3 x - 1 - 2 x - 3) - (x - 4)^{2} \\ = (5 x + 2) (x - 4) - (x - 4)^{2} \\ = (x - 4) [(5 x + 2) - (x - 4)] \\ = (x - 4) [4 x + 6] \\ = 2 (x - 4) (2 x + 3) \end{matrix}$

Identités remarquables

Développement

Quotient