Problem 3

Let

f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0

and

g(x)=c_{n+1}x^{n+1}+c_nx^n+\cdots+c_0

be non-zero real polynomials such that

g(x)=(x+r)f(x)

for some real

r

. If

a=\max(|a_n|,\ldots,|a_0|)

and

c=\max(|c_{n+1}|,\ldots,|c_0|)

, prove that

\frac{a}{c}\le n+1

Step 4 of 5: Use backward induction when |r| is less than 1

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0<|r|<1\Longrightarrow |a_n|\le c,\quad |a_{n-1}|<2c,\quad |a_{n-k}|<(k+1)c

Detailed analysis

For $0<|r|<1$ , $|a_n|=|c_{n+1}|\le c$ and $|a_{n-1}|=|c_n-ra_n|<c+c=2c$ . If $|a_{n-k}|\le(k+1)c$ , then $|a_{n-k-1}|=|c_{n-k}-ra_{n-k}|<c+(k+1)c=(k+2)c$ . Hence every coefficient satisfies $|a_j|\le(n+1)c$ .