Section outline

  • عام

    طي الكل توسيع الكل

  • développer et reduire les expressions

    • 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. \( (x - 2)(2x - 1)(5 - x) \)}

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

  • Développer les expressions

    • 1. Développer les expressions suivantes :

      - a. \( 2(3x - 1)(2 - x) \)

      \[
      2(3x - 1)(2 - x) = (6x - 2)(2 - x)
      \]
      \[
      = 12x - 6x^2 - 4 + 2x
      \]
      \[
      = -6x^2 + 14x - 4
      \]

      - b. \( (2x + 3)^2 \)

      \[
      (2x + 3)^2 = 4x^2 + 12x + 9
      \]

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

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

      - d. \( (5x - 6)^2 \)

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

      2. Factoriser les expressions suivantes :

      - a. \( (x + 1)(1 - x) - (x + 1)(2x + 1) \)

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

      - b. \( 3(2x - 2) + (x + 1)(1 - x) \)

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

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

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

      - d. \( 12x^2 - 6x + (2x - 1)(5 - 2x) \)

      \[
      12x^2 - 6x + (2x - 1)(5 - 2x) = 6x(2x - 1) + (2x - 1)(5 - 2x)
      \]
      \[
      = (2x - 1)[6x + (5 - 2x)]
      \]
      \[
      = (2x - 1)(4x + 5)
      \]

      Effectuer les factorisations suivantes :

      a. \( (3x + 1)(2 - 2x) - (5 - 4x)(x - 1) \)}

      On remarque l'égalité : \( 2 - 2x = -2(x - 1) \)

      \[
      (3x + 1)(2 - 2x) - (5 - 4x)(x - 1) = (3x + 1)[-2(x - 1)] - (5 - 4x)(x - 1)
      \]
      \[
      = (x - 1)[-2(3x + 1) - (5 - 4x)]
      \]
      \[
      = (x - 1)(-6x - 2 - 5 + 4x) = (x - 1)(-2x - 7)
      \]

      b. \( (2 - 3x)(3 + 2x) + (3x + 2)(-6x - 9) \)}

      On a l'égalité : \( -6x - 9 = -3(2x + 3) \)

      \[
      (2 - 3x)(3 + 2x) + (3x + 2)(-6x - 9) = (2 - 3x)(3 + 2x) + (3x + 2)[-3(2x + 3)]
      \]
      \[
      = (2x + 3)[(2 - 3x) - 3(3x + 2)]
      \]
      \[
      = (2x + 3)(2 - 3x - 9x - 6)
      \]
      \[
      = (2x + 3)(-12x - 4)
      \]
      \[
      = -4(2x + 3)(3x + 1)
      \]

      c. \( (6x + 2)(2x + 3) + (9x + 3)^2 \)}

      On a : \( 6x + 2 = 2(3x + 1) \) ; \( 9x + 3 = 3(3x + 1) \)

      \[
      (6x + 2)(2x + 3) + (9x + 3)^2 = [2(3x + 1)(2x + 3)] + [3(3x + 1)]^2
      \]
      \[
      = (3x + 1)[2(2x + 3) + 9(3x + 1)]
      \]
      \[
      = (3x + 1)(4x + 6 + 27x + 9)
      \]
      \[
      = (3x + 1)(31x + 15)
      \]

      d. \( (3x + 3)^2 - (x + 2)(5x + 4) \)}

      \[
      (3x + 3)^2 - (x + 2)(5x + 4) = (9x^2 + 18x + 9) - (5x^2 + 4x + 10x + 8)
      \]
      \[
      = 9x^2 + 18x + 9 - 5x^2 - 14x - 8
      \]
      \[
      = 4x^2 + 4x + 1 = (2x + 1)^2
      \]

    • Effectuer les factorisations suivantes : 

      a. \( (3x + 1)(2 - 2x) - (5 - 4x)(x - 1) \)}

      On remarque l'égalité : \( 2 - 2x = -2(x - 1) \)

      \[
      (3x + 1)(2 - 2x) - (5 - 4x)(x - 1) = (3x + 1)[-2(x - 1)] - (5 - 4x)(x - 1)
      \]
      \[
      = (x - 1)[-2(3x + 1) - (5 - 4x)]
      \]
      \[
      = (x - 1)(-6x - 2 - 5 + 4x) = (x - 1)(-2x - 7)
      \]

      b. \( (2 - 3x)(3 + 2x) + (3x + 2)(-6x - 9) \)}

      On a l'égalité : \( -6x - 9 = -3(2x + 3) \)

      \[
      (2 - 3x)(3 + 2x) + (3x + 2)(-6x - 9) = (2 - 3x)(3 + 2x) + (3x + 2)[-3(2x + 3)]
      \]
      \[
      = (2x + 3)[(2 - 3x) - 3(3x + 2)]
      \]
      \[
      = (2x + 3)(2 - 3x - 9x - 6)
      \]
      \[
      = (2x + 3)(-12x - 4)
      \]
      \[
      = -4(2x + 3)(3x + 1)
      \]

      c. \( (6x + 2)(2x + 3) + (9x + 3)^2 \)}

      On a : \( 6x + 2 = 2(3x + 1) \) ; \( 9x + 3 = 3(3x + 1) \)

      \[
      (6x + 2)(2x + 3) + (9x + 3)^2 = [2(3x + 1)](2x + 3) + [3(3x + 1)]^2
      \]
      \[
      = (3x + 1)[2(2x + 3) + 9(3x + 1)]
      \]
      \[
      = (3x + 1)(4x + 6 + 27x + 9)
      \]
      \[
      = (3x + 1)(31x + 15)
      \]

      d. \( (3x + 3)^2 - (x + 2)(5x + 4) \)}

      \[
      (3x + 3)^2 - (x + 2)(5x + 4) = (9x^2 + 18x + 9) - (5x^2 + 4x + 10x + 8)
      \]
      \[
      = 9x^2 + 18x + 9 - 5x^2 - 14x - 8
      \]
      \[
      = 4x^2 + 4x + 1 = (2x + 1)^2
      \]

    • Factoriser les expressions suivantes : 

      - a. \( (x-1)(2x+1) - (2x-2)(5-2x) \)

      \[
      (x-1)(2x+1) - (2x-2)(5-2x) = (x-1)(2x+1) - 2(x-1)(5-2x)
      \]
      \[
      = (x-1)[(2x+1) - 2(5-2x)]
      \]
      \[
      = (x-1)(2x+1 - 10 + 4x)
      \]
      \[
      = (x-1)(6x - 9)
      \]
      \[
      = 3(x-1)(2x - 3)
      \]

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

      \[
      (2+x)(3-x) + (5-2x)(3-x) = [(2+x) + (5-2x)](3-x)
      \]
      \[
      = (2 + x + 5 - 2x)(3 - x)
      \]
      \[
      = (7 - x)(3 - x)
      \]

      - c. \( 3(4+2x) - (3+x)(10+5x) \)

      \[
      3(4+2x) - (3+x)(10+5x) = 3[2(2+x)] - (3+x)[5(2+x)]
      \]
      \[
      = 6(2+x) - 5(3+x)(2+x)
      \]
      \[
      = [6 - 5(3+x)](2+x)
      \]
      \[
      = (6 - 15 - 5x)(2+x)
      \]
      \[
      = 5(-9 - 5x)(2+x)
      \]
      \[
      = -(5x + 9)(x + 2)
      \]

      - d. \( (2-x)(3x-4) + \left(2-\frac{3}{2}x\right)(2x+3) \)

      \[
      (2-x)(3x-4) + \left(2-\frac{3}{2}x\right)(2x+3) = (2-x)(3x-4) + \left[-\frac{1}{2}(-4+3x)\right](2x+3)
      \]
      \[
      = (2-x)(3x-4) - \frac{1}{2}(-4+3x)(2x+3)
      \]
      \[
      = \left[(2-x) - \frac{1}{2}(2x+3)\right](3x-4)
      \]
      \[
      = \left(2 - x - x - \frac{3}{2}\right)(3x-4)
      \]
      \[
      = \left(\frac{1}{2} - 2x\right)(3x-4)
      \]

      On pouvait trouver également \(\left(2-\frac{3}{2}x\right)(4x-1)\) qui est une expression égale.

      - e. \( (2x+1)^2 - 4(2-3x)^2 \)

      \[
      (2x+1)^2 - 4(2-3x)^2 = (2x+1)^2 - [2(2-3x)]^2
      \]
      \[
      = [(2x+1) + 2(2-3x)][(2x+1) - 2(2-3x)]
      \]
      \[
      = (2x+1 + 4 - 6x)(2x+1 - 4 + 6x)
      \]
      \[
      = (5 - 4x)(8x - 3)
      \]

      - f. \( 18x^2 - 24x + 8 + (3x-2)(2-x) \)

      \[
      18x^2 - 24x + 8 + (3x-2)(2-x) = 2(9x^2 - 12x + 4) + (3x-2)(2-x)
      \]
      \[
      = 2(3x-2)^2 + (3x-2)(2-x)
      \]
      \[
      = (3x-2)\left[2(3x-2) + (2-x)\right]
      \]
      \[
      = (3x-2)(6x - 4 + 2 - x)
      \]
      \[
      = (3x-2)(5x - 2)
      \]

  • Développer chacune des expressions

    • Développer chacune des expressions suivantes :}

      a. \( x(x - 3) - x^2 \) 

       b. \( (6x + 1)^2 + (12x + 2)(3 - 3x) \) 

      c. \( (x + 1)^2 - (x - 1)^2 \) 

      d. \( 2(3x - 1)(2 - x) \) 

      e. \( (2x + 3)^2 \) 

      f. \( (3x - 2)(3x + 2) \)

      j. \( (5x - 6)^2 \)

  • Factoriser, si possible, les expressions

    • Factoriser, si possible, les expressions suivantes : 

      - a. \( 3(x+1) + 5(x-1) \)}

      - b. \( 3(x+1) - 2(x+1) - (x-1) \)}

      - c. \( x^2 + 5 \times (-5) \)}

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

      - e. \( x(x+1) - x^2 + 5x \)}

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

      - g. \( 3(x-2) + (x+2)(2-x) \)}

      - h. \( (6x-9)(x+1) - (4x-6) \)}

      -i. \( (2x + 3)(1 - x) + (4x + 6)^2 \)}

      -j. \( (3 - 9x)^2 + 3(3x - 1) \)}

      -k. \( (5x + 1)(2x - 4) + (3x - 6)^2 \)}