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$$|z^3+3z^2-2z+1|\leq|z^3+3z^2-2z|+|1|\leq|z^3+3z^2|+|-2z|+|1|\leq|z^3|+|3z^2|+|-2z|+|1|$$

[applying the triangle inequality three times in succession]

$$=|z|^3+3|z|^2+|-2||z|+1$$

[applying (8) ]

$$=|z|^3+3|z|^2+2|z|+1<2^3+3\cdot 2^2+2\cdot 2+1=25$$

EDIT: Just saw Samy's post. I don't have the book but, based on that picture, it looks more likely a typo in that the ref to (9) should be to (10), rather than you miscopying it.

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Samy_A

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May well be a typo. He is referring to the following:

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Thank you very much.

$$|z^3+3z^2-2z+1|\leq|z^3+3z^2-2z|+|1|\leq|z^3+3z^2|+|-2z|+|1|\leq|z^3|+|3z^2|+|-2z|+|1|$$

[applying the triangle inequality three times in succession]

$$=|z|^3+3|z|^2+|-2||z|+1$$

[applying (8) ]

$$=|z|^3+3|z|^2+2|z|+1<2^3+3\cdot 2^2+2\cdot 2+1=25$$

EDIT: Just saw Samy's post. I don't have the book but, based on that picture, it looks more likely a typo in that the ref to (9) should be to (10), rather than you miscopying it.

- #5

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Thank you very much.May well be a typo. He is referring to the following:

View attachment 99135

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