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##### If $F$ is the field of rational numbers, find necessary and sufficient conditions on $a$ and $b$ so that the splitting field of $x^{3}+a x+b$ has degree exactly 3 over $F$.
If $F$ is the field of rational numbers, find necessary and sufficient conditions on $a$ and $b$ so that the splitting field of $x^{3}+a x+b$ has degree exactly 3 over $F$....
##### Definition. Let $T$ be a lincar operator on a finite-dimensional vector spacc $\mathrm{V}$, and let $x$ be a nonzero vector in $\mathrm{V}$. The polynomial $p(t)$ is called a $T$ -annihilator of $x$ if $p(t)$ is a monic polynomial of least degroo for which $p(T)(x)=0$ Let $T$ be a linear operator on a finite-dimensional vector space $V$, and let $x$ be a nonzero vector in $V$. Prove the following results. (a) The vector $x$ lins a mique T-amililator. (b) The T-annihilator of $x$ divides any poly
Definition. Let $T$ be a lincar operator on a finite-dimensional vector spacc $\mathrm{V}$, and let $x$ be a nonzero vector in $\mathrm{V}$. The polynomial $p(t)$ is called a $T$ -annihilator of $x$ if $p(t)$ is a monic polynomial of least degroo for which $p(T)(x)=0$ Let $T$ be a linear operator on...
##### 234Fm emits 6 successive alpha particles and then two beta particles. What is the product nucleus at the end of this sequence? Write the equations for the successive reactions. (2 points)
234Fm emits 6 successive alpha particles and then two beta particles. What is the product nucleus at the end of this sequence? Write the equations for the successive reactions. (2 points)...
##### Complete parts a-c for each quadratic function.a. Find the $y$ -intercept, the equation of the axis of symmetry, and the $x$ -coordinate of the vertex.b. Make a table of values that includes the vertex.c. Use this information to graph the function.$$f(x)=-0.25 x^{2}-3 x$$
Complete parts a-c for each quadratic function. a. Find the $y$ -intercept, the equation of the axis of symmetry, and the $x$ -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$f(x)=-0.25 x^{2}-3 x$$...
##### If $\mathrm{h}(x)=2(x+1)$ and $\mathrm{h}(x)=\mathrm{f}(\mathrm{g}(x)),$ what are possible expressions for $\mathrm{f}(x)$ and for $\mathrm{g}(x) ?$
If $\mathrm{h}(x)=2(x+1)$ and $\mathrm{h}(x)=\mathrm{f}(\mathrm{g}(x)),$ what are possible expressions for $\mathrm{f}(x)$ and for $\mathrm{g}(x) ?$...
##### Consider nitration of 2-methylnaphthalene at the 1-, 6- and 8-positions. Draw the all resonance Structures for intermediates for each reaction. Based, on the stability relevant intermediates, predict which should be main product of nitration:CH3
Consider nitration of 2-methylnaphthalene at the 1-, 6- and 8-positions. Draw the all resonance Structures for intermediates for each reaction. Based, on the stability relevant intermediates, predict which should be main product of nitration: CH3...
##### Stokes radiation is_________ than the incomingphoton while it occurs at a ___________Rayleigh radiation.
Stokes radiation is _________ than the incoming photon while it occurs at a ___________ Rayleigh radiation....