Answer Key 10.5
- [latex]\text{let }u=x^2[/latex]
[latex]\therefore u^2-5u+4=0[/latex]
[latex]\text{factors to }(u-4)(u-1)=0[/latex]
[latex]\text{replace }u: (x^2-4)(x^2-1)=0 \\[/latex]
[latex](x-2)(x+2)(x-1)(x+1)=0[/latex]
[latex]x=\pm 2, \pm 1[/latex] - [latex]\text{let }u=y^2[/latex]
[latex]\therefore u^2-9y+20=0[/latex]
[latex]\text{factors to }(u-5)(u-4)=0[/latex]
[latex]\text{replace }u: (y^2-5)(y^2-4)=0 \\[/latex]
[latex]\begin{array}{ll} y^2-5=0\hspace{0.25in}&(y-2)(y+2)=0 \\ y^2=5&y=\pm 2 \\ y=\pm \sqrt{5}& \end{array}[/latex] - [latex]u=m^2[/latex]
[latex]\therefore u^2-7u-8=0[/latex]
[latex](u-8)(u+1)=0[/latex]
[latex](m^2-8)(m^2+1)=0 \\[/latex]
[latex](m+\sqrt{8})(m-\sqrt{8})(m^2+1)=0[/latex]
[latex]m=\pm \sqrt{8}\text{ or }\pm 2\sqrt{2}[/latex]
[latex]m^2+1\text{ has 2 non-real solutions}[/latex] - [latex]u=y^2[/latex]
[latex]\therefore u^2-29y+100=0[/latex]
[latex](u-25)(u-4)=0[/latex]
[latex](y^2-25)(y^2-4)=0 \\[/latex]
[latex](y-5)(y+5)(y-2)(y+2)=0[/latex]
[latex]y=\pm 5, \pm 2[/latex] - [latex]\text{let }u=a^2[/latex]
[latex]\therefore u^2-50u+49=0[/latex]
[latex](u-49)(u-1)=0[/latex]
[latex](a^2-49)(a^2-1)=0 \\[/latex]
[latex](a-7)(a+7)(a-1)(a+1)=0[/latex]
[latex]a=\pm 7, \pm1[/latex] - [latex]\text{let }u=b^2[/latex]
[latex]\therefore u^2-10u+9=0[/latex]
[latex](u-9)(u-1)=0[/latex]
[latex](b^2-9)(b^2-1)=0 \\[/latex]
[latex](b-3)(b+3)(b-1)(b+1)=0[/latex]
[latex]b=\pm 3, \pm 1[/latex] - [latex]x^4-20x^2+64=0[/latex]
[latex]\text{let }u=x^2[/latex]
[latex]\therefore u^2-20u+64=0[/latex]
[latex](u-16)(u-4)=0[/latex]
[latex](x^2-16)(x^2-4)=0 \\[/latex]
[latex](x-4)(x+4)(x-2)(x+2)=0[/latex]
[latex]x=\pm 4, \pm 2[/latex] - [latex]6z^6-z^3-12=0[/latex]
[latex]\text{let }u=z^3[/latex]
[latex]\therefore 6u^2-u-12=0[/latex]
[latex](3u+4)(2u-3)=0[/latex]
[latex](3z^3+4)(2z^3-3)=0\\[/latex]
[latex]\begin{array}{ll} 3z^3+4=0\hspace{0.25in}&2z^3-3=0 \\ 3z^3=-4&2z^3=3 \\ \\ z^3=-\dfrac{4}{3}&z^3=\dfrac{3}{2} \\ \\ z=\sqrt[3]{-\dfrac{4}{3}}&z=\sqrt[3]{\dfrac{3}{2}} \end{array}[/latex] - [latex]z^6-19z^3-216=0[/latex]
[latex]\text{let }u=z^3[/latex]
[latex]\therefore u^2-19u-216=0[/latex]
[latex](u-27)(u+8)=0[/latex]
[latex](z^3-27)(z^3+8)=0 \\[/latex]
[latex](z-3)(z^2+3z+9)(z+2)(z^2-2z+4)=0[/latex]
[latex]z=3, -2[/latex]
[latex]2\text{ non-real solutions each for the 2nd and 4th factors}[/latex] - [latex]\text{let }u=x^3[/latex]
[latex]\therefore u^2-35u+216=0[/latex]
[latex](u-27)(u-8)=0[/latex]
[latex](x^3-27)(x^3-8)=0[/latex]
[latex](x-3)(x^2+3x+9)(x-2)(x^2+2x+4)[/latex]
[latex]x=2, 3[/latex]
[latex]2\text{ non-real solutions each for the 2nd and 4th factors}[/latex]
