[latex]a)\ -5x^4 + 3x^3 + 14x^2 = 0 \ x^2(-5x^2+3x+14)=0\ x^2=0 vee -5x^2+3x+14=0\ \ Delta = 3^2-4cdot (-5)cdot 14=289\ x=frac{-3-17}{-10}=2\ x=frac{-3+17}{-10}=-frac{7}{5}\ \ x={-frac{7}{5}, 0, 2}\ \ \ b)\ 4x^4- 5x^2 + 1 = 0\ t=x^2\ 4t^2-5t+1=0\ Delta=5^2-4cdot 4=9\ \ t=frac{5-3}{8}=frac{1}{4}\ t=frac{5+3}{8}=1\ \ x=sqrt{frac{1}{4}}=frac{1}{2} vee -frac{1}{2}\ x=sqrt{1}=1 vee -1\ \ x={-1, -frac{1}{2}, frac{1}{2}, 1}\ [/latex] [latex]c)\ 2x^5 + 5x^3 - 12x = 0\ x(2x^4+5x^2-12)=0\ x=0 vee 2x^4+5x^2-12=0\ \ 2t^2+5t-12=0\ Delta=5^2-4cdot 2 cdot (-12)=121\ \ t=frac{-5-11}{4}=-4\ t=frac{-5+11}{4}=frac{6}{4}\ \ x=sqrt{frac{6}{4}}=frac{sqrt{6}}{2} vee -frac{sqrt{6}}{2}\ \ x={-frac{sqrt{6}}{2}, 0, frac{sqrt{6}}{2}}\ \ d)\ 2x^7 + x^4 + x = 0\ x(2x^6+x^3+1)=0\ x=0 vee 2x^6+x^3+1=0\ \ t=x^3\ 2t^2+t+1=0\ Delta<0\ \ x={0}\ \ \ [/latex] [latex]e)\ 6x^3 + 6x^2-3x-3 =0 \ 6x^2(x+1)-3(x+1)=0\ (6x^2-3)(x+1)=0\ 3(3x^2-1)(x+1)=0\ (sqrt{3}x-1)(sqrt{3}x+1)(x+1)=0\ sqrt{3}x-1=0 =vee sqrt{3}x+1=0 vee x+1=0\ x=frac{sqrt{3}}{3} vee x=-frac{sqrt{3}}{3} vee x=-1\ \ x={-1, -frac{sqrt{3}}{3}, frac{sqrt{3}}{3}}\ \ \ f)\ 2x^5-18x^3 + 2x^2=18\ 2x^5-18x^3+2x^2-18=0\ 2x^3(x^2-9)+2(x^2-9)=0\ 2(x^3+1)(x^2-9)=0\ (x+1)(x^2-x+1)(x-3)(x+3)=0\ x=-1 vee x=3 vee x=-3\ \ x={-3, -1, 3} [/latex]
