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 5 of 5: Take the maximum and finish

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a\le(n+1)c\Longrightarrow \frac{a}{c}\le n+1

Detailed analysis

The preceding cases all establish $a\le(n+1)c$ . Since $c>0$ for the non-zero polynomial $g$ , dividing by $c$ gives $\frac{a}{c}\le n+1$ , equivalently $\frac{a}{c}\le n+1$ under the source's normalized coefficient notation.