Prove that all the roots of [latex]z^7-5 z^3+12=0[/latex] lie between the circles |z| = 1 and |z| = 2.

Consider the circle [latex]C_1:|z|=1[/latex]. Let [latex]f(z)=12, g(z)=z^7-5 z^3[/latex]. On [latex]C_1[/latex] we have [latex]|g(z)|=\left|z^7-5 z^3\right| \leq\left|z^7\right|+\left|5 z^3\right| \leq 6<12=|f(z)|[/latex] Hence, by Rouché's theorem, [latex]f(z)+g(z)=z^7-5 z^3+12[/latex] has the same number of zeros inside |z|=1 as [latex]f(z)=12[/latex], i.e., there are no zeros inside [latex]C_1[/latex]. Consider the circlc [latex]C_2:|z|=2[/latex]. Let [latex]f(z)=z^7, g(z)=12-5 z^3[/latex]. On [latex]C_2[/latex] we have [latex]|g(z)|=\left|12-5 z^3\right| \leq|12|+\left|5 z^3\right| \leq 60<2^7=|f(z)|[/latex] Hence, by Rouché's theorem, [latex]f(z)+g(z)=z^7-5 z^3+12[/latex] has the same number of zeros inside |z|=2 as [latex]f(z)=z^7[/latex], i.e., all the zeros are inside [latex]C_2[/latex]. Hence, all the roots lie inside |z|=2 but outside |z|=1, as required.

Question 5.20: Prove that all the roots of z^7 - 5z³ + 12 = 0 lie between t...

Schaum’s Outline of Complex Variables

Prove that all the roots of $z^7-5 z^3+12=0$ lie between the circles |z| = 1 and |z| = 2.

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Consider the circle $C_1:|z|=1$ . Let $f(z)=12, g(z)=z^7-5 z^3$ . On $C_1$ we have

|g(z)|=\left|z^7-5 z^3\right| \leq\left|z^7\right|+\left|5 z^3\right| \leq 6<12=|f(z)|

Hence, by Rouché’s theorem, $f(z)+g(z)=z^7-5 z^3+12$ has the same number of zeros inside |z|=1 as $f(z)=12$ , i.e., there are no zeros inside $C_1$ .
Consider the circlc $C_2:|z|=2$ . Let $f(z)=z^7, g(z)=12-5 z^3$ . On $C_2$ we have

|g(z)|=\left|12-5 z^3\right| \leq|12|+\left|5 z^3\right| \leq 60<2^7=|f(z)|

Hence, by Rouché’s theorem, $f(z)+g(z)=z^7-5 z^3+12$ has the same number of zeros inside |z|=2 as $f(z)=z^7$ , i.e., all the zeros are inside $C_2$ .
Hence, all the roots lie inside |z|=2 but outside |z|=1, as required.