Factoring a Polynomial. The Relation Between Coefficients and Roots

Determine $a$ and $b$ so that $x^{4}+3x^{2}+ax+b$ is divisible without remainder by $x^{2}-2ax+2$.

Show that for integers $m$, $n$, and $p$, the polynomial $x^{3m}+x^{3n+1}+x^{3p+2}$ is divisible by $x^{2}+x+1$.

Prove that $(x+y)^{n}-x^{n}-y^{n}$ is divisible by $x^{2}+xy+y^{2}$ for $n=6m+5$, and by $(x^{2}+xy+y^{2})^{2}$ for $n=6m+1$.

The equation $3x^{4}-5x^{3}+3x^{2}+4x-2=0$ has the root $1+i$. Find the remaining roots.

Prove that a double root of the equation $\varphi (x)^{2}+\psi (x)^{2}=0$ satisfies the equation $\varphi '(x)^{2}+\psi '(x)^{2}=0$, if the polynomials $\varphi (x)$ and $\psi (x)$ are relatively prime.

Prove that a root of multiplicity $n$ of the equation

is a root of multiplicity $n-1$ of the equation $F\left(u'(x), v'(x)\right)=0$. Here $u(x)$, $v(x)$ are relatively prime polynomials and $F(u,v)$ is a homogeneous function having no repeated linear factors.

Prove that a polynomial $f(x)$ remains positive for all real $x$ if and only if

where $\alpha_{0}, \alpha_{1}, \alpha_{2}, \ldots , \alpha_{n}$ are certain complex numbers.

Prove the identity:

Prove the identity:

Prove the identity:

Prove the identity:

Prove the identity:

Prove the identity:

Prove the identity:

Prove the identity:

for odd or even $n$ respectively.

Prove the identity:

Prove the identity:

Prove the identity:

Factor the polynomial $(x+1)^{n}+(x-1)^{n}$.

Factor the polynomial $x^{2n}+C_{2n}^{2}x^{2n-2}(x^{2}-1)+C_{2n}^{4}x^{2n-4}(x^{2}-1)^{2}+\ldots +(x^{2}-1)^{n}$.

Factor the polynomial $x^{2n+1}+C_{2n+1}^{2}x^{2n-1}(x^{2}-1)+C_{2n+1}^{4}x^{2n-3}(x^{2}-1)^{2}+\ldots +x(x^{2}-1)^{n}$.

Evaluate the product $(x_{1}^{2}+1)(x_{2}^{2}+1)\ldots (x_{n}^{2}+1)$, where $x_{1}, x_{2}, \ldots , x_{n}$ are the roots of the polynomial $x^{n}+a_{1}x^{n-1}+\ldots +a_{n}$ ($a_{1}, a_{2}, \ldots , a_{n}$ real).

The roots of the polynomial $f(x)=a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n}$ are related by $x_{s}+x_{n-s}=m$, $s=1, 2, \ldots , n$. Prove the identity: $f(x)=(-1)^{n}f(m-x)$.

The roots of a polynomial $f(x)$ of degree $2n$ are related by $x_{s}+x_{n+s}=2m$, $s=1, 2, \ldots , n$. Prove that one of the roots of $f'(x)$ equals $m$, and the rest are related pairwise by the formulas $y_{s}+y_{n+s-1}=2m$, $s=1, 2, \ldots , n-1$.

The equation $a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n}=0$, where $a_{0}a_{n} \neq 0$, has $k+1$ terms. Show that it cannot have roots of multiplicity higher than $k$.

Is the polynomial

divisible by $(x-1)^{4}$?

Find the relation between the coefficients of the polynomial $a_{0}x^{3}+a_{1}x^{2}+a_{2}x+a_{3}=0$, if its roots satisfy the relation $x_{1}x_{2}+x_{2}x_{3}=2x_{1}x_{3}$.

Find the relation between $p$, $q$, and $r$ for which the roots of the equation $x^{3}+px^{2}+qx+r=0$ form a geometric progression.

For what $\lambda$ and $\mu$ do the roots of the equation $x^{4}+2x^{3}-21x^{2}+\lambda x+\mu =0$ form an arithmetic progression, and what progression specifically?

For what $p$, $q$, $\alpha$ ($0 < \alpha < 4$) does the equation $x^{4}+px^{\alpha }+q=0$ have a triple root?

The remainders upon dividing a polynomial by $(x-a)$, $(x-b)$, $(x-c)$ are $A$, $B$, $C$. Find the remainder upon dividing by $(x-a)(x-b)(x-c)$.

The roots of the polynomial $x^{3}+x^{2}-2$ are $1$, $\alpha$, and $\beta$. Find the second-degree polynomial $\varphi (x)$ for which $\varphi (1)=1$, $\varphi (\alpha )=\beta$, $\varphi (\beta )=\alpha$, and prove that $\varphi (\varphi (x))-x$ is divisible by $x^{3}+x^{2}-2$.

Find a seventh-degree polynomial $f(x)$ such that $f(x)+1$ is divisible by $(x-1)^{4}$, and $f(x)-1$ is divisible by $(x+1)^{4}$.

Find a polynomial $f(x)$ of degree $2n-1$ such that $f(x)+1$ is divisible by $(x-1)^{n}$, and $f(x)-1$ by $(x+1)^{n}$.

Find a polynomial of degree $n$ that, upon division by $(x-a_{1}), (x-a_{2}), \ldots , (x-a_{n})$, gives remainders $A_{1}, A_{2}, \ldots , A_{n}$.

Prove the validity of Euler's identity

where $\varphi (x)=(x-x_{1})(x-x_{2})\ldots (x-x_{n})$, the numbers $x_{1}, x_{2}, \ldots , x_{m}$ are distinct, and $\omega (x)$ is any polynomial of degree not exceeding $m-2$.

In the equality

both sides can be expanded in powers of $z$ if the numbers $\left|\lambda z\right|$, $\left|\lambda^{-1}z\right|$, and $\left|(\lambda +\lambda^{-1})z-z^{2}\right|$ are less than unity. By equating, on both sides, the coefficients of like powers of $z$, prove the identity

Setting $\lambda =\cos \varphi +i\sin \varphi$ in the identity of the preceding problem, obtain, for the Chebyshev polynomial $T_{n}(x)=2^{-(n-1)}\cos (n\arccos x)$, the explicit expression

Taking into account the behavior of the polynomial $T_{n}(x)$ on the interval $-1 \leqslant x \leqslant 1$, prove that any other polynomial $x^{n}+a_{1}x^{n-1}+\ldots +a_{n}$ with real coefficients cannot remain, in absolute value, everywhere less than $2^{-(n-1)}$ — that is, that the Chebyshev polynomial deviates least from zero on the interval $(-1, 1)$.

$F(x)=f(x)\varphi (x)^{k}$, where $f(x)$ and $\varphi (x)$ are relatively prime polynomials with no repeated roots. Setting $F(x)=P(x)Q(x)$, $F'(x)=P(x)R(x)$, where $Q(x)$ and $R(x)$ are relatively prime, prove that $\varphi (x)$ is the greatest common divisor of the polynomials $Q(x)$ and $R(x)-kQ'(x)$.