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 3 of 5: Use forward induction when

|r|\ge1

Previous step Next step

|r|\ge1\Longrightarrow |a_0|\le c,\quad |a_1|\le2c,\quad |a_k|\le(k+1)c

Detailed analysis

For $|r|\ge1$ , $|a_0|=|c_0/r|\le c$ . Also $|a_1|=|(c_1-a_0)/r|\le|c_1|+|a_0|\le2c$ . If $|a_k|\le(k+1)c$ , then $|a_{k+1}|=|(c_{k+1}-a_k)/r|\le|c_{k+1}|+|a_k|\le(k+2)c$ . Thus $|a_k|\le(k+1)c\le(n+1)c$ for every $k$ .