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 2 of 5: Handle the case r=0

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r=0\Longrightarrow a=c\Longrightarrow \frac{a}{c}=1\le n+1

Detailed analysis

If $r=0$ , then $c_0=0$ and $c_k=a_{k-1}$ for $1\le k\le n$ , while $c_{n+1}=a_n$ . Thus the maxima agree: $a=c$ , and because $f$ is non-zero, $a=c>0$ . Hence $\frac{a}{c}=1\le n+1$ .