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