Let p(z) be a polynomial of degree n ge 1 and let z0 in mathbbC. Show that there is a polynomial

Name: Let p(z) be a polynomial of degree n ge 1 and let z0 in mathbbC. Show that there is a polynomial h(z) of degree n - 1 such that p(z) = (z - z0)h(z) + p(z0). In particular, if p(z0) = 0, then p(z) = (z - z0)h(z).
Uploaded: 2023-01-20T00:01:35-08:00
Duration: 7 min 4 s
Channel: Vincenzo Zaccaro
Description: Let p(z) be a polynomial of degree n ge 1 and let z0 in mathbbC. Show that there is a polynomial h(z) of degree n - 1 such that p(z) = (z - z0)h(z) + p(z0). In particular, if p(z0) = 0, then p(z) = (z - z0)h(z).

Question

Vincenzo Zaccaro · Accepted Answer

Okay, so we are going to prove the claim in our exercise by induction. So induction. Okay, well,

Question

Let $p(z)$ be a polynomial of degree $n ge 1$ and let $z_0 in mathbb{C}$. Show that there is a polynomial $h(z)$ of degree $n - 1$ such that $p(z) = (z - z_0)h(z) + p(z_0)$. In particular, if $p(z_0) = 0$, then $p(z) = (z - z_0)h(z)$.

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