Question

descompuneti in factori
2x la a doua +4x+2=?
x la a treia -4x la a doua +4x=?
x la a patra +6x la a treia +9x la a doua =?
3x la a treia -24x la a doua +48x=?
-2x la a doua -20x-50=?
2x la a doua +0,08-0,8x=?
x la a patra -x la a treia +x la a doua supra 4=?
0,03x la a treia +1,8x+27=?

Answer 1

2x² + 4x + 2 = (x√2 + √2)(x√2 + √2) 

x³ - 4x² + 4x = x × (x² - 4x + 4) = x × (x - 2)(x - 2) 

x⁴ + 6x³ + 9x² = (x² + 3x)(x² + 3x) 

3x³ - 24x² + 48x = 3x × (x² - 8x + 16) = 3x × (x - 4)(x - 4) 

-2x² - 20x - 50 = -2 × (x² + 10 + 25) = - 2 × (x + 5)(x + 5) 

2x² + 0,08 - 0,8x = 2 × ( x² + 0,04 - 0,4) = 2 × ( x - 0,2)(x - 0,2) 

x⁴ - x³ + x²/4 = (x² - x/2)(x - x/2) =

     (x-√(x/2)) (x + √(x/2) (x-√(x/2)) (x + √(x/2))

Formule de calcul prescurtat:

• (a + b)² = a² + 2ab + b²

• (a – b)² = a² – 2ab + b²

• (a + b)(a – b) = a² – b²

• (a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc

• (a – b + c)² = a² + b² + c² – 2ab + 2ac – 2bc

• (a + b – c)² = a² + b² + c² + 2ab – 2ac – 2bc

• (a – b – c)² = a² + b² + c² – 2ab – 2ac – 2bc

• (a + b)³ = a³ + 3a²b + 3ab² + b³

• (a – b)³ = a³ – 3a²b + 3ab² – b³

• a³ + b³ = (a + b)(a² – ab + b²)

• a³ – b³ = (a – b)(a² + ab + b²)

Answer 2

$2x^2+4x+2= 2x^2+2x+2x+2 = \\ =2x(x+1)+2(x+1) = (x+1)(2x+2) = \\ = 2(x+1)(x+1) = 2(x+1)^2 \\ \text{sau:}\\ 2x^2+4x+2 = 2(x^2+2x+1) = 2(x+1)^2 \\ \\ x^3 -4x^2+4x= x(x^2-4x+4) = x(x-2)^2 \\ \\ x^4+6x^3+9x^2= x^2(x^2+6x+9) = x^2(x+3)^2 = \Big[x(x+3)\Big]^2\\ \\ 3x^3-24x^2+48x= 3x(x^2-8x+16) = 3x(x-4)^2 \\ \\ -2x^2-20x-50= -2(x^2+10x+25) = -2(x+5)^2 \\ \\2x^2+0,08-0,8x= 2(x^2+0,04-0,4x) = \\ =2(x^2-0,4x+0,04) = 2(x-0,2)^2$

$x^4-x^3 +\dfrac{x^2}{4} =x^2\Big(x^2-x+\dfrac{1}{4}\Big) = x^2\Big(x-\dfrac{1}{2}\Big)^2 = \\ = \left[x\Big(x-\dfrac{1}{2}\Big)\right]^2\\ \\ \\ 0,03x^2 +1,8x+27 = 3\cdot (0,01x^2+0,6x+9) = \\ =\dfrac{3}{100}\cdot(x^2+60x+900) = \dfrac{3}{100}\cdot (x+30)^2 = 0,03\cdot (x+30)^2$