Section I.6.4

Polynomials with real coefficients

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Problem I.6.4.1

Let $f(X)=a_{0} X^{n}+a_{1} X^{n-1}+\ldots +a_{n}$ be a real polynomial of degree $n$ . Show that the values of the upper bounds for the positive roots of the polynomials $f(X), X^{n} f(1 / X), f(-X), X^{n} f(-1 / X)$ give lower and upper bounds for both the positive and the negative roots of the polynomial $f(X)$ .

Problem I.6.4.2

In the notation of Exercise 1, let $a_{0}>0, m$ be the smallest index for which $a_{m}<0, B$ the maximum among the absolute values of the negative coefficients. Show that

c \leqslant 1+\sqrt[m]{B / a_{0}}

for every positive real root of the polynomial $f(X)$ .

Problem I.6.4.3

Let $P$ be a field of characteristic zero, $a \in P$ . For any polynomial $f \in P[X]$ of degree $n$ , the formula (Taylor's formula) holds

f(X)=f(a)+\frac{f^{\prime }(a)}{1 !}(X-a)+\frac{f^{\prime \prime }(a)}{2 !}(X-a)^{2}+\ldots +\frac{f^{(n)}(a)}{n !}(X-a)^{n}.

Problem I.6.4.4

Show that if $f(a)>0, f^{\prime }(a)>0, \ldots , f^{(n)}(a)>0$ for a real polynomial $f(X)$ of degree $n$ with positive leading coefficient $a_{0}$ , then $f(c)=0, c>0 \Longrightarrow c<a$

Problem I.6.4.5

Using Descartes' rule of signs, find the sign of the discriminant of the polynomials $X^{5}-X^{2}+1, X^{3}-6 X-9$ (see the remark at the end of item 1).

Problem I.6.4.6

Can the polynomials $X^{5}-X-1, X^{3}+a X+b \in \mathbb {Q}{}[X]$ have common complex roots? Recall (see Exercise 11 from § 1) that the polynomial $X^{5}-X-1$ is irreducible over $\mathbb {Q}$ .

Problem I.6.4.7

Show that the roots of the polynomial $f(X)=X^{5}+u X^{4}+v X^{3}+w \in \mathbb {R}{}[X]$ with nonzero constant term $w \neq 0$ cannot all be real.

Problem I.6.4.8

Any polynomial $f(X)$ with $f(x) \geqslant 0$ for all $x \in \mathbb {R}$ can be represented in the form

f(X)=g(X)^{2}+h(X)^{2}

where $g, h \in \mathbb {R}{}[X]$

Problem I.6.4.9

Derive on your own the stability criterion for polynomials of degrees 3 and 4. For $n=4$ , write it as a system of inequalities:

a_{1}>0, \quad a_{4}>0, \quad a_{1} a_{2}>0, \quad a_{3}\left(a_{1} a_{2}-a_{3}\right)>a_{1}^{2} a_{4}

Problem I.6.4.10

Does the polynomial $f(X)$ standing in the numerator of the irreducible rational fraction

\frac{f(X)}{g(X)}=\frac{3}{X+2}+\frac{1}{(X-1)^{2}}-\frac{2}{X-1}+\frac{X-3}{X^{2}+1}

have real roots?

Problem I.6.4.11

Show that all three roots of the polynomial $f(z)=z^{3}-7 z-7$ , irreducible over $\mathbb {Q}$ , are real and lie in the interval $]-2,4[$ . Compute the positive root by Newton's method to within the third decimal place.

Problem I.6.4.12

Relying on Rouché's theorem (Theorem 4), show that the polynomial $f(z)=z^{5}+5 z^{2}-3$ has two roots in the unit disk and three roots in the annulus between the circles $\left|z\right|=1$ and $\left|z\right|=2$ .