Zadanie 55 z załącznika. Rozkładanie wielomianu na czynniki, zadanie trudniejsze.

[latex]W_a(x) = x^3(x^2-7)^2-36x = xig(x^2(x^2-7)^2-36ig) =\=xig((x(x^2-7))^2-36ig)=x(x(x^2-7)-6)(x(x^2-7)+6)=\=x(x^3-7x-6)(x^3-7x+6)=x(x+1)(x^2-x-6)(x-1)cdot\cdot(x^2+x-6)\sqrt{Delta_1} =sqrt{1+24} = sqrt{25} = 5 \sqrt{Delta_2} = sqrt{1+24} =5 \W_a(x)=x(x-1)(x+1)(x+2)(x-3)(x+3)(x-2)[/latex] [latex]W_b(x) = x^3(x^2+2)^2-9x = xig((x(x^2+2))^2-9ig)=\=x(x(x^2+2)-3)(x(x^2+2)+3) = x(x^3+2x-3)(x^3+2x+3)=\=x(x-1)(x^2+x+3)(x+1)(x^2-x+3)\Delta_1 = 1-4cdot1cdot3 extless 0 \ Delta_2 = 1-4cdot1cdot3 extless 0[/latex] [latex]W_c(x) = (x^2-2)^4 - 4x^4 = ig((x^2-2)^2-2x^2ig)ig((x^2-2)^2+2x^2ig)=\=(x^2-2-sqrt{2}x)(x^2-2+sqrt{2}x)ig((x^2-2)^2+2x^2ig)\sqrt{Delta_1} = sqrt{2+8} = sqrt{10} \ sqrt{Delta_2} = sqrt{10}\W_c(x) = (x-frac{sqrt{2}-sqrt{10}}{2})(x-frac{sqrt{2}+sqrt{10}}{2})(x+frac{sqrt{2}+sqrt{10}}{2})(x+frac{sqrt{2}-sqrt{10}}{2})cdot\cdotig(x^4-2x^2+4ig)[/latex] [latex]\(x^4-2x^2+4) = (t^2-2t+4) qquadqquad Delta = 4-4cdot4 extless 0\(x^4-2x^2+4) = (x^2+sqrt{6}x+2)(x^2-sqrt{6}x+2)\W_c(x) = (x-frac{sqrt{2}-sqrt{10}}{2})(x-frac{sqrt{2}+sqrt{10}}{2})(x+frac{sqrt{2}+sqrt{10}}{2})(x+frac{sqrt{2}-sqrt{10}}{2})cdot\cdot(x^2+sqrt{6}x+2)(x^2-sqrt{6}x+2)[/latex] [latex]W_d(x) = (x^2+9)^4 - 16x^4 = ig((x^2+9)^2-4x^2ig)ig((x^2+9)^2+4x^2ig)=\=(x^2+9-2x)(x^2+9+2x)ig(x^4+18x^2+81+4x^2ig)\Delta_1 = 4-4cdot9cdot1 extless 0\Delta_2 = 4-4cdot9cdot1 extless 0\ (x^4+22x^2+81) = (t^2+22t+81) qquadqquad Delta = 22^2-4cdot81 = 160\(x^4+22x^2+81) = (x^2 - frac{-22-4sqrt{10}}{2})(x^2-frac{-22+4sqrt{10}}{2})\W_d(x) = (x^2+9-2x)(x^2+9+2x)(x^2 +11+2sqrt{10})(x^2+11-2sqrt{10})[/latex]

a) [latex]\x^3(x^2-7)^2-36x=x*[x^2(x-7)^2-36]= \ \x*[x^2(x^4-14x^2+49)-36]= \ \x(x^6-14x^4+49x^2-36)= ... W(-1)=W(1)=0 \ \x*[x^4(x^2-1)-13x^2(x^2-1)+36(x^2-1)]= \ \x(x^2-1)(x^4-13x^2+36)=... W(-2)=W(2)=0 \ \x(x^2-1)*[x^2(x^2-4)-9(x^2-4)]= \ \x(x^2-1)(x^2-4)(x^2-9)= \ \x(x+1)(x-1)(x+2)(x-2)(x+3)(x-3)[/latex] b) [latex]\x^3(x^2+2)^2-9x=x*[x^2(x^4+4x^2+4)-9]= \ \x(x^6+4x^4+4x^2-9)=... W(-1)=W(1)=0 \ \x*[x^4(x^2-1)+5x^2(x^2-1)+9(x^2-1)]= \ \x(x^2-1)(x^4+5x^2+9)= \ \x(x^2-1)*[(x^4+6x^2+9)-x^2]=x(x^2-1)*[(x^2+3)^2-x^2]= \ \x(x+1)(x-1)(x^2-x+3)(x^2+x+3)[/latex] c) [latex]\(x^2-2)^4-4x^4= \ \.[(x^2-2)^2-2x^2]*[(x^2-2)^2+2x^2]= \ \(x^4-4x^2+4-2x^2)*(x^4-4x^2+4+2x^2)= \ \.[(x^4-6x^2+9)-5]*[(x^4+4x^2+4)-6x^2]= \ \(x^2-3-sqrt{5} )(x^2-3+ sqrt{5})* \ \(x^2+2-x sqrt{6} )(x^2+2+x sqrt{6} )= \ \(x^2+2-x sqrt{6} )(x^2+2+x sqrt{6} )* \ \(x- sqrt{3+ sqrt{5} } )(x+ sqrt{3+ sqrt{5} } )(x- sqrt{3- sqrt{5} })(x+ sqrt{3+ sqrt{5} })[/latex] d) [latex]\(x^2+9)^4-16x^4=[(x^2+9)^2-4x^2]*[(x^2+9)^2+4x^2]= \ \(x^2-2x+9)(x^2+2x+9)(x^4+22x^2+81)= \ \(x^2-2x+9)(x^2+2x+9)*[(x^4+22x^2+121)-40]= \ \(x^2-2x+9)(x^2+2x+9)*[(x^2+11)^2-40]= \ \(x^2+11-2 sqrt{10} )(x^2+11+2 sqrt{10} )* \ \(x^2-2x+9)(x^2+2x+9)[/latex]