Lien entre développer et factoriser

A
$3x + 21 = 3x + 3 \times 7 = 3(x + 7)$

B
$4x - x^2 = x(4 - x)$

C
$-5x + 20 = -5x + 5 \times 4 = 5(-x + 4)$

D
$5x^2 - 8x = x(5x - 8)$

E
$(x - 1)(2x + 3) - (x - 1)(2 - x)$
$= (x - 1)[(2x + 3) - (2 - x)]$
$= (x - 1)(2x + 3 - 2 + x)$
$= (x - 1)(3x + 1)$

F
$(2x + 1)^2 + (2x + 1)(x + 3)$
$= (2x + 1)[(2x + 1) + (x + 3)]$
$= (2x + 1)(2x + 1 + x + 3)$
$= (2x + 1)(3x + 4)$

G
$(5x - 2)(2x + 7) - (5x - 2)$
$= (5x - 2)[(2x + 7) - 1]$
$= (5x - 2)(2x + 6)$
$= 2(5x - 2)(x + 3)$

H
$7x - 49 + 14x^2 = 14x^2 + 7x - 49$
$= 7(2x^2 + x - 7)$

I
$9x^2 + 12x + 4$
$= (3x)^2 + 2 \times 3x \times 2 + 2^2$
$= (3x + 2)^2$

J
$(2x - 7)(x + 4) - (2x - 7)(4x + 1)$
$= (2x - 7)[(x + 4) - (4x + 1)]$
$= (2x - 7)(x + 4 - 4x - 1)$
$= (2x - 7)(-3x + 3)$
$= 3(2x - 7)(-x + 1)$

K
$(4x - 1)^2 + (2x - 5)(4x - 1)$
$= (4x - 1)[(4x - 1) + (2x - 5)]$
$= (4x - 1)(4x - 1 + 2x - 5)$
$= (4x - 1)(6x - 6)$
$= 6(4x - 1)(x - 1)$

L
$(x + 7)(3x - 1) + 7x + 49$
$= (x + 7)(3x - 1) + 7(x + 7)$
$= (x + 7)(3x - 1 + 7)$
$= (x + 7)(3x + 6)$
$= 3(x + 7)(x + 2)$

M
$16x^2 - 81 = (4x)^2 - 9^2 = (4x - 9)(4x + 9)$

N
$49x^2 - \dfrac14 = (7x)^2 - \left(\dfrac12\right)^2$
$= \left(7x - \dfrac12\right)\left(7x + \dfrac12\right)$

O
$9x^2 + 30x + 25$
$= (3x)^2 + 2 \times 3x \times 5 + 5^2$
$= (3x + 5)^2$

P
$(2x + 3)^2 - 49 = (2x + 3)^2 - 7^2$
$= [(2x + 3) - 7][(2x + 3) + 7]$
$= (2x - 4)(2x + 10)$
$= 2(x - 2) \times 2(x + 5)$
$= 4(x - 2)(x + 5)$

Q
$(4x - 1)^2 - (2x + 3)^2$
$= [(4x - 1) - (2x + 3)][(4x - 1) + (2x + 3)]$
$= (4x - 1 - 2x - 3)(4x - 1 + 2x + 3)$
$= (2x - 4)(6x + 2)$
$= 2(x - 2)\times 2(3x + 1)$
$= 4(x - 2)(3x + 1)$

R
$x^3 - 16x = x(x^2 - 16)$
$= x(x - 4)(x + 4)$

S
$25x^2 - 1 - (4x - 3)(5x + 1)$
$= (5x)^2 - 1 - (4x - 3)(5x + 1)$
$= (5x - 1)(5x + 1) - (4x - 3)(5x + 1)$
$= (5x + 1)[(5x - 1) - (4x - 3)]$
$= (5x + 1)(x + 2)$

T
$x^2 + 8x + 16 = x^2 + 2 \times x \times 4 + 4^2 = (x + 4)^2$

U
$4x^2 - 4x + 1 = (2x)^2 - 2 \times 2x \times 1 + 1^2 = (2x - 1)^2$

V
$x^2 - 64 = x^2 - 8^2 = (x - 8)(x + 8)$

W
$x^2 + x + 0,25 = x^2 + 2 \times x \times 0,5 + 0,5^2 = (x + 0,5)^2$

X
$100x^2 - 1,000x + 2,500 = (10x)^2 - 2 \times 10x \times 50 + 50^2$
$= (10x - 50)^2$
$= [10(x - 5)]^2$
$= 100(x - 5)^2$

Y
$16x^2 - \dfrac{81}{4} = (4x)^2 - \left(\dfrac92\right)^2$
$= \left(4x - \dfrac92\right)\left(4x + \dfrac92\right)$

Z
$x^2 - 7 = x^2 - (\sqrt{7})^2$
$= (x - \sqrt{7})(x + \sqrt{7})$

A'
$2x^2 + 2 = 2(x^2 + 1)$

B'
$(3x - 1)^2 - (x + 2)^2$
$= [(3x - 1) - (x + 2)][(3x - 1) + (x + 2)]$
$= (3x - 1 - x - 2)(3x - 1 + x + 2)$
$= (2x - 3)(4x + 1)$