Question

Let z0 be a zero of the polynomial P(z)=a0 +a1z+a2z2 +···+anzn of degree n (n ≥...

Let z0 be a zero of the polynomial

P(z)=a0 +a1z+a2z2 +···+anzn of degree n (n ≥ 1). Show in the following way that

P(z) = (z − z0)Q(z) where Q(z) is a polynomial of degree n − 1.

(an ̸=0)

(k=2,3,...).

(a) Verify that
zk−zk=(z−z)(zk−1+zk−2z +···+zzk−2+zk−1)

00000 (b) Use the factorization in part (a) to show that

P(z) − P(z0) = (z − z0)Q(z)
where Q(z) is a polynomial of degree n − 1, and deduce the desired result from this.

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