Resol y=(x-1)(x-2)^2(x-3)^3(x-4)

$$y = ( x - 1 ) ( x - 2 ) ^ { 2 } ( x - 3 ) ^ { 3 } ( x - 4 )$$

Resoleu y

$$y=\left(x-4\right)\left(x-1\right)\left(x-2\right)^{2}\left(x-3\right)^{3}$$

Passos de solució

Utilitzeu el teorema del binomi \left(a-b\right)^{2}=a^{2}-2ab+b^{2} per desenvolupar \left(x-2\right)^{2}.

$$y=\left(x-1\right)\left(x^{2}-4x+4\right)\left(x-3\right)^{3}\left(x-4\right)$$

Utilitzeu el teorema del binomi \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} per desenvolupar \left(x-3\right)^{3}.

$$y=\left(x-1\right)\left(x^{2}-4x+4\right)\left(x^{3}-9x^{2}+27x-27\right)\left(x-4\right)$$

Utilitzeu la propietat distributiva per multiplicar x-1 per x^{2}-4x+4 i combinar-los com termes.

$$y=\left(x^{3}-5x^{2}+8x-4\right)\left(x^{3}-9x^{2}+27x-27\right)\left(x-4\right)$$

Utilitzeu la propietat distributiva per multiplicar x^{3}-5x^{2}+8x-4 per x^{3}-9x^{2}+27x-27 i combinar-los com termes.

$$y=\left(x^{6}-14x^{5}+80x^{4}-238x^{3}+387x^{2}-324x+108\right)\left(x-4\right)$$

Utilitzeu la propietat distributiva per multiplicar x^{6}-14x^{5}+80x^{4}-238x^{3}+387x^{2}-324x+108 per x-4 i combinar-los com termes.

$$y=x^{7}-18x^{6}+136x^{5}-558x^{4}+1339x^{3}-1872x^{2}+1404x-432$$