Exercice 1:

Développer et donner la forme réduite des expressions ci-dessous :

-a. \( (3x + 2)(5 - 2x) \) 

\[
(3x + 2)(5 - 2x) = 15x - 6x^2 + 10 - 4x
\]
\[
= -6x^2 + 11x + 10
\]

-b. \( (x - 1)(3x^2 - 2) \) 

\[
(x - 1)(3x^2 - 2) = 3x^3 - 2x - 3x^2 + 2
\]
\[
= 3x^3 - 3x^2 - 2x + 2
\]

-c. \( 2(3 - 2x)x - 2(x - 2) \)

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

-d. \( [2 + 2(x - 5)](x - 1) \)

\[
[2 + 2(x - 5)](x - 1) = (2 + 2x - 10)(x - 1)
\]
\[
= (2x - 8)(x - 1) = 2x^2 - 2x - 8x + 8
\]
\[
= 2x^2 - 10x + 8
\]

-e. \( (5x + 1)[2(x - 1) - 5x] \)

\[
(5x + 1)[2(x - 1) - 5x] = (5x + 1)(2x - 2 - 5x)
\]
\[
= (5x + 1)(-3x - 2) = -15x^2 - 10x - 3x - 2
\]
\[
= -15x^2 - 13x - 2
\]

Développer les expressions suivantes :

- a.  \( (2x + 1)(3 - x) \)

\[
(2x + 1)(3 - x) = 6x - 2x^2 + 3 - x
\]
\[
= -2x^2 + 5x + 3
\]

- b. \( (5 - 2x)(3 - x) - 3(3 - 2x) \)}

\[
(5 - 2x)(3 - x) - 3(3 - 2x) = 15 - 5x - 6x + 2x^2 - 9 + 6x
\]
\[
= 2x^2 - 5x + 6
\]

- c. \( (x + 1)^2 + (2x - 1)^2 \)}

\[
(x + 1)^2 + (2x - 1)^2 = (x^2 + 2x + 1) + (4x^2 - 4x + 1)
\]
\[
= 5x^2 - 2x + 2
\]

- d. \( (x - 2)(2x - 1)(5 - x) \)}

\[
(x - 2)(2x - 1)(5 - x) = (2x^2 - x - 4x + 2)(5 - x)
\]
\[
= (2x^2 - 5x + 2)(5 - x)
\]
\[
= 10x^2 - 2x^3 - 25x + 5x^2 + 10 - 2x
\]
\[
= -2x^3 + 15x^2 - 27x + 10
\]

Développer les expressions suivantes : 

- a. \( (2x + 1)(3 - x) \) 

\[
(2x + 1)(3 - x) = 6x - 2x^2 + 3 - x
\]
\[
= -2x^2 + 5x + 3
\]

- b. \( (5 - 2x)(3 - x) - 3(3 - 2x) \)

\[
(5 - 2x)(3 - x) - 3(3 - 2x) = 15 - 5x - 6x + 2x^2 - 9 + 6x
\]
\[
= 2x^2 - 5x + 6
\]

- c. \( (x + 1)^2 + (2x - 1)^2 \)}

\[
(x + 1)^2 + (2x - 1)^2 = (x^2 + 2x + 1) + (4x^2 - 4x + 1)
\]
\[
= 5x^2 - 2x + 2
\]

- d. \( (x - 2)(2x - 1)(5 - x) \)}

\[
(x - 2)(2x - 1)(5 - x) = (2x^2 - x - 4x + 2)(5 - x)
\]
\[
= (2x^2 - 5x + 2)(5 - x)
\]
\[
= 10x^2 - 2x^3 - 25x + 5x^2 + 10 - 2x
\]
\[
= -2x^3 + 15x^2 - 27x + 10
\]