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Question Number 46378 by rahul 19 last updated on 24/Oct/18

$$Factorise:$$$$3x^4+6x^3+8x^2-2x-3=0.$$

Answered by MJS last updated on 24/Oct/18

$$\mathrm{factor}1=3$$$$x^4+2x^3+\frac{8}{3}{x}^2-\frac{2}{3}x-1=0$$$$(x^2+ax+b)(x^2+cx+d)=0$$$$\mathrm{by}\mathrm{comparing}\mathrm{the}\mathrm{constants}\mathrm{we}\mathrm{get}4\mathrm{equations}$$$$\left(1\right)a+c-2=0$$$$\left(2\right)ac+b+d-\frac{8}{3}=0$$$$\left(3\right)ad+bc+\frac{2}{3}=0$$$$\left(4\right)bd+1=0$$$$\mathrm{solving}\left(1\right)\mathrm{for}a,\left(2\right)\mathrm{for}b,\left(3\right)\mathrm{for}d\mathrm{we}\mathrm{get}$$$$\left(4\right)9c^6-54c^5+156c^4-364c^3+280c^2-176c=0$$$$\Rightarrow c=0,a=2,b=3,d=-\frac{1}{3}$$$$\mathrm{so}\mathrm{we}\mathrm{have}$$$$3(x^2+2x+3)(x^2-\frac{1}{3})=0$$$$\mathrm{solving}\mathrm{the}\mathrm{square}\mathrm{factors}\mathrm{we}\mathrm{get}$$$$3(x+1-\mathrm{i}\sqrt{2})(x+1+\mathrm{i}\sqrt{2})(x-\frac{\sqrt{3}}{3})(x+\frac{\sqrt{3}}{3})=0$$

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