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 1 of 5: Expand the product and record coefficient recurrences

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c_{n+1}=a_n,\qquad c_k=a_{k-1}+ra_k\ (1\le k\le n),\qquad c_0=ra_0

Detailed analysis

Comparing coefficients in $g(x)=(x+r)f(x)$ gives $c_{n+1}=a_n$ , $c_k=a_{k-1}+ra_k$ for $1\le k\le n$ , and $c_0=ra_0$ . By definition every coefficient of $g$ has absolute value at most $c$ .