Consider the polynomial P(z) = ao+a+ az+...+ anzn of degree n, where ao, a1,..., an are...

  

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Consider the polynomial P(z) = ao+a₁²+ a₂z²+...+ anzn of degree n, where ao, a1,..., an are complex valued constants with an # 0. Show that for any real number c with 0 < c < 1 there exists a large enough R > 0 such that 1 |P(z)| 1 (1-c)|an|Rn¹ < whenever |z|> R. Note that we used this in the proof of the Fundamental Theorem of Algebra in class. Hint: first show that lao + a1z + + an-12-1| ≤ can||z|" whenever |z| > R as long as R is sufficiently large. Then show that |P(z)| ≥ |anz”|— ao + a₁2 + +an-12-11. ... Activate W Consider the polynomial P(z) = ao+a₁²+ a₂z²+...+ anzn of degree n, where ao, a1,..., an are complex valued constants with an # 0. Show that for any real number c with 0 < c < 1 there exists a large enough R > 0 such that 1 |P(z)| 1 (1-c)|an|Rn¹ < whenever |z|> R. Note that we used this in the proof of the Fundamental Theorem of Algebra in class. Hint: first show that lao + a1z + + an-12-1| ≤ can||z|" whenever |z| > R as long as R is sufficiently large. Then show that |P(z)| ≥ |anz”|— ao + a₁2 + +an-12-11. ... Activate W