- Let a and b be the two complex roots of the polynomial 2x³+2x+1. Find, with proof, the smallest degree monic polynomial with integer coefficients with a/b as a root.
Answer by Josh Swanson
Since you haven’t gotten an answer yet, I’ll mention that Wolfram Alpha can at least compute the minimal polynomial in question: x^6 + 3x^5 + 10x^4 + 15x^3 + 10x^2 + 3x + 1. Naively plugging in awful expressions for the roots in would prove that the ratio is a root, and there are deterministic algorithms for proving the irreducibility of polynomials over Q, but this is a terrible and backwards approach.
- I was put in AP Physics B for next year because of my high math grades in middle school. Also, I have a strong interest in math and science. I am wondering how much work I will need to put towards this.
Answer by Imre
You don’t need to put real work into it. You like it, so it’s mainly fun rather than work. Have fun acquiring some interesting, nice knowledge!
- I play the B♭ Clarinet, and i bought a box of all bad reeds except for 2 of them!! Is there a way that i can make them good again? I bought the exact brand (Rico) and size (3) that i usually use, but one 2 of them were good! Also, how do i make it easier to switch octaves? I have a hard time going from A to B.
Answer by Just Me
For starters, don’t buy Rico again. They’re cheap and inconsistent as you have probably already figured out. Second, there are many ways to modify your reeds. I always begin by checking to see if they are warped. Check this by placing it on a flat surface and gently pressing on either side as if to rock it back and forth. If it rocks, it’s warped and you need to sand down the flat side until it stays put. Also, if you have a reed knife, you can slightly shave away at the wings. This will help with the resistance. Work on that part slowly and keep testing it in between shavings so you don’t overdo it. Good luck!
An easy way to keep teeth whitened is to brush after consuming something that tends to stain teeth. Common culprits are coffee, sodas, wines, teas, berries, and sweets. If you brush immediately after consuming these items, you prevent the stain from setting into your teeth. This will help you to keep your teeth white without having to resort to expensive treatments. There are plenty of simple home remedies you can use to get whiter teeth naturally. An excellent method is using baking soda. Use the baking soda box from your kitchen or purchase one of the many commercially available baking soda toothpastes.Teeth whitening can be bothersome and expensive. There are a number of different methods that can help you whiten Carry On Examining Through
When you chose a credit card because of the reward of miles or cash that are offered, make sure to understand how the reward program works before you agree to the credit card terms. Check to see if there is an expiration date on the reward and decide if you can take advantage of them before they expire. Do not forget to factor in your credit cards when creating a budget. You should already be budgeting your income, so just include your credit cards in your existing budget. A credit card's available balance should not be considered extra money. Have a set amount you are happy to spend monthly using this card and stick to it. Stick to that, and pay it off each month. It is a good idea to contact credit card companies and Carry On Examining Through
You can defeat your struggle with depression by changing the messages that you repeat in your mind silently. Don't let negativity flow through your thoughts; instead turn those negative ideas into positive ones. The more you are able to pound this ideal into your brain, the more real it becomes to your perception.Change your negative behaviors that contribute towards your downward slide into depression. Replace your negative habits with positive, self-affirming habits that will help you feel good about yourself. If a situation upsets you, remember that blaming yourself and succumbing to depression is not the answer. Don't get sabotaged with problems, be assertive. To make sure that you are getting the best treatment Carry On Examining Through
My old apartment burning down was one of the most devastating moments of my life. I was able to get out with some of my belongings, but most of my stuff was burned in the fire. The fire was caused by some electrical wiring problems. My apartment wasn't the only one that burned, as many of my neighbors had their apartments burned as well. Some people didn't have anywhere to go after the fire. Luckily I had my parents to fall back on. While living at home, my parents told me about a website called http://thestratfordapartments.com.My parents were pretty helpful in finding me a new apartment to live in. It's almost as if they wanted me out of the house again so they could be alone. Maybe it's just my imagination, but at Carry On Examining Through
With luck, this past week I was able to snag a unit at one of the best apart complexes in town; the Waterford. Yeah, I bet you're reading this right now and burning up with envy, shocked that I was actually able to get it right from under your noses! Ha! It's true - I can't believe I was able to get it myself. I spent months on the waiting list, going back in every week and speaking with the property managers whenever I could to make sure that they knew I was a familiar face around. Is that cheating? I hope not! Sometimes when you want something badly enough you have to do everything that you can to ensure that you're the one to get it, especially when it's limited like these units are. I've got to admit; I love the Carry On Examining Through
Know who your audience is, and reach out for them. If you are aware that your readers spend time on different social media sites, than you want to be doing that as well. Your readers come to your site because they feel they have certain interests in common with you. When you show up at the same social media sites that they frequent, you affirm that belief.Don't use an excessive number of keywords. Although effective keyword choices play a big role in improving your blog's visibility, it is definitely a field where quality trumps quantity. Search engines are always tightening their criteria when it comes to good content. You can actually be penalized by the search engines for including too many keywords on a page. Instead, Carry On Examining Through
It's important to do extensive research before signing up for health insurance if you suffer from a pre-existing medical condition. You might not even be eligible for health insurance from some companies if you have one of these conditions; others charge people with chronic or serious conditions far more each month than people who don't have these conditions. Start your research by reading the article below, and you'll be equipped to have the best health care experience that you can. When you buy health insurance, you should become familiar with your local regulations and laws regarding it first. For instance, does your legislation offer any protection to people with preexisting medical issues? Knowing these laws will Carry On Examining Through
When it comes to whitening your teeth, you have many excellent products to choose from. See what a few different stores have to offer before deciding on a product. Then, pick the option that most appeals to you and use it regularly. Remember that directions for usage vary greatly from one brand to the next, so pay close attention to the fine print. Taking fluoride supplements can also strengthen teeth. Fluoride might be useful if your teeth aren't as white as you want or if you are having trouble with your gums. You may get yellow spots on your teeth, however, if you take in too much of it. Too much fluoride is not good for you, so try to keep your intake low. You may want to have out-dated mercury fillings replaced. Carry On Examining Through
Some business tank because of a poor reputation. Failing to value your customers will cause word to spread. Keeping that in mind, here are some suggestions about how to do things the right way to gain customers, earn profits and see your company grow.Always remember that there is no way to please everyone. There are many business owners that get so discouraged when they get negative feedback that they allow it to mold the shape of their business. While it doesn't feel good to see negative things about you, do not allow it to get you down. Competition is high in almost any type of industry that you can think of. A customer gives his business to the company with the best reputation because he is almost guaranteed to be Carry On Examining Through
You should have all medications you take written down. If you use different pharmacies, this will allow the pharmacist to cross check all medications. You can have the pharmacist go over your medications to see if any of them interact.There are a couple of things that will happen as you age. One, everyone dies eventually. The second is that your body will age eventually. Luckily, you have found a great source for useful tips that can help you as you age. You should have a savings account for health issues in addition to your retirement account. While staying insured during this time is a must, you still need a monetary buffer for a worst case scenario. Think of going to the grocery store like going to the pharmacy, and Carry On Examining Through
I sort of expected that it would be boring and I probably would have tried to get out of it if I could have. However this is one of those things where you get invited to something and it is important that you show up in order to keep the peace, in this case with my girlfriend and her family. It turned out to be pretty interesting, although I did not really need to learn what smoked salmon quiche tastes like to be honest. They probably spent a huge fortune on the catering and probably thought it would impress people if they got the sort of food that rich people eat. Of course I would have liked to had a steak and so forth. However the interesting part was in the actions of the newly related by law people. It was not surprising Carry On Examining Through
Trust your instincts when it comes to deals that seem too good to be true online. Shopping over the Internet is much different than your local brick-and-mortar store in that you can't just go in and speak to the manager. Once you've submitted your credit card or bank information, you can't take it back, so read all the fine print before buying.I want to save money when I shop online, but I don't know how? You are not alone, my friend, as most people who buy on the internet have no idea how to do it cheaply. That is why we have written this article to be full of secrets you need to know.You might be hesitant to start shopping online, but there's really no reason to be. This is something that more and more people are doing, Carry On Examining Through
When you send mobile marketing messages, send along a discount offer or a promo code. When customers see these offers, they may feel that they want to look at your site to see what you are offering.Mobile marketing has many different definitions associated with it. When you look at the mobile marketing techniques of several businesses, you will notice contrasts in both definition and execution. Essentially, it means advertising on cell phones or other mobile devices to promote your business. This article will give you some tips to make that kind of mobile marketing work for your business.Mobile marketing is extremely perplexing. Businesses are all very different, and so are their marketing campaigns. A method that works Carry On Examining Through
The CPAP machine is often prescribed for sleep apnea. This machine provides positive, continuous airway pressure. Use it by covering your airways with the included mask. Once you connect the mask to the CPAP machine with a short hose, the smooth flow of pressurized air will help you breathe freely through the night. CPAP machines work really well. However, learning how to sleep at night with a mask on takes some time to become accustomed to. If you wear a CPAP for your sleep apnea symptoms, but are still bothered by your mouth opening when you try to sleep, get a chin strap. The chin strap holds your chin so that you cannot sleep with your mouth open; it is a tiny piece of fabric. Use this device to solve this annoying Carry On Examining Through
Before selling any of your gold, check to see what the current sales price is. This will help ensure that you are getting the amount of money you deserve. There are many reputable sites available online that will help you on coming up with a fair price for your gold. You can easily invest in gold if you do your research first. Now that you know some helpful information about gold, you should feel comfortable going into this market. Follow this advice to reap good returns on your gold investments.If you are headed to a gold party, keep your expectations low. While you may walk out with a few hundred dollars, you are not going to get rich. You also won't get the full value of your jewelry. Go and have fun, but do not sell Carry On Examining Through
Your business may want to comprehend – Need Advice On Organic Gardening? Look No Further Than This Article!
Use equal parts of green and dried plant material in your compost pile. When you pull weeds from your garden, throw them in the compost. The same goes for vegetable trimmings and grass clippings. These are considered green materials. Sawdust, straw, cardboard, paper and wood pulp are all examples of dried plant material. However, stay away from meats, manure from carnivores, charcoal and other potentially harmful substances.One way to help your organic garden thrive is to leave an undeveloped area that is conducive to the wildlife around your area. Wildlife can help the plants in your garden to thrive, as insects support plant reproduction, while the excrement of many species contains nutrients which can help to fertilize Carry On Examining Through
Perhaps an internal nasal dilator would reduce your snoring. For most people, snoring through the nose is not common, but it does still happen. Nasal dilators are made to fit into your nasal passages and keep them open. If this is the cause of your snoring, this will likely be a very effective treatment. Excessive snoring could be a sign of sleep apnea, which is a more serious condition. If you experience symptoms such as gasping for breath at night, extreme fatigue during the day, difficulty focusing or recalling information, or your partner tells you that you sometimes stop breathing during sleep, make an appointment to see your doctor as soon as you can. Sleep apnea not only messes with your life on a daily basis, Carry On Examining Through
Find ways to share your crafting skill with others. Teaching your children how to join you in doing a craft is a great way to pass down your special skill. It is also a great way to spend time with your family and to make memories that will last a lifetime. Anyone who participates regularly in arts and crafts pursuits can attest to the relaxation and fun they can bring. Those interested in adopting one or more such hobbies for themselves should learn all they can about potential pastimes. The piece below can help readers do just that and find a great deal of enjoyment.Doing arts and crafts products doesn't have to be expensive. There are several ways that you can cut costs if you are creative and recycle commonly used Carry On Examining Through
When conducting a mobile marketing campaign, you must provide a link to your website for smart phone and tablet users, and also your mobile website to accommodate users of older cell phones. Remember only a small portion of mobile phones can use flash, so if your standard website has flash on it, link to version mobile only.You should remember that Internet navigation is a lot harder on a mobile device. If your marketing website is easy to use, you will get a better response from people using mobile devices. Don't worry too much about how your site looks on a home computer. The important thing is keeping your site user friendly for both users. Make your mobile marketing advertisements easy to share, and they will be Carry On Examining Through
Whenever you have free time, work on passing the ball. This is a fundamental concept, but also a key one. You should practice all you can. A great way to practice this is setting up targets for you to hit. Put them at various distances, and work on your long and short passes. One thing about soccer is you're going to end up in a world of hurt if you get distracted. You must stay focused, and you must learn about things you can control vs things you can't control. You may think you're focused, but if you're focusing on things you can't control, you're actually quite distracted. Direct free kicks result from kicking another player, tripping another player, charging a player,jumping at a player, striking an opponent, pushing Carry On Examining Through
As soon as the carpets are cleaned, check the completed work. If certain areas seem dirty call the company immediately. The longer you wait, the better the chances of a company denying your claim. During vacuuming, divide the floor into sections to make vacuuming easier. This means you can see what areas you already did so you do not double track. Divide your room into sections to save both time and effort.If the carpet cleaner offers warranties for their work, remember that for the future. For example, before you apply additional treatments to your carpet, inform your carpet cleaning company about your intentions. You may accidentally void your warranty without meaning to.You want to provide ample space for the cleaning Carry On Examining Through
Buy a good coffee grinder. When you grind your own beans immediately before brewing them, your coffee will maintain its aroma and flavor. Many coffee grinders have variable settings in determining the coarseness of your coffee grinds that you wind up with. You can also find a coffee maker that has it's own grinder if you desire fewer appliances. Wait until the entire pot of coffee is finished brewing before pouring your first cup even if your machine has a pause function. There are a few machines that let you pour a cup prior to the brew finishing, but this degrades the quality of the entire pot. You should use one equipped with a timer instead. Your coffee will be ready for you when you arise from your slumber.This Carry On Examining Through
Any credit cards that have balances over 50% of your limit should be paid off until they are less than 50% of your limit. When balances are over 50%, your credit rating goes down significantly, so try to either spread out your debt or, ideally, pay off your credit cards.If you have a poor credit history and can't qualify for a credit card, get a secured card. The account is funded in advance, which provides the lender with a level of insurance against unpaid debts. This is why secured cards are often easier to get. If you use it correctly, a new card can help you fix your credit.When you have poor credit, a quick way to provide a little boost is to contact your financial institutions to request a lowering of your credit Carry On Examining Through
If you are considering a long-haired cat breed then you should also consider the extra maintenance required. Even though that nice coat of beautiful silky fur looks great, it can leave a mess on your floors, furniture and clothing. Don't get such a cat unless you are prepared for the extra commitment they take over regular cats. Not to mention cats that have long hair are more prone to getting issues with hairballs. If you absolutely must bathe a cat, set everything up in advance. Set up two tubs large enough to dunk a cat. Fill them with comfortably warm water. Set your open bottle of cat shampoo, a heavy towel, and a secure pet carrier close by. Wear heavy rubber gloves. Grasp your cat by the scruff of the neck. Dunk, Carry On Examining Through
Here is a way of treating acne that is a little bit off the wall. You want to mix cut up strawberries with sour cream and make sure it is blended well. Then apply the mixture to your face like a beauty mask. This mixture will cool your face and clear up problem areas.Many people think that sun exposure can help acne, but this isn't always true. Sometimes, the sun and heat can make your skin produce excess oil, which leads to more blemishes. And of course there is also the enhancement of wrinkles, and the dangers of skin cancer. When you see that huge zit, most people are tempted to pop it. Instead, do not touch it at all! By popping the zit, you release the white blood cells that had been attacking the bacteria on your Carry On Examining Through
One of the most important things that you can do to prevent asthma attacks is to identify your triggers. You need to think about maintaining a journal that can assist you in determining potential triggers of asthma. Talk to your doctor about these results. After you figure them out, try to stay away from your triggers, and take them out of your environment if you can.Watch for signs of a serious asthma attack in order to know whether you should rush to the hospital with your child. Lips and fingernails that are grey-blue in color and a lack of response to medication increase are both signs that the attack is a serious one. Your child can have a hard time speaking. Learn as much as possible about asthma. The more you Carry On Examining Through
Keep a top 10 list on your blog of your best work so users can see what your blog focuses on. By doing this, your viewers will easily be able to determine of your content is that which they seek, and subsequently your search engine listings shall be enhanced. You should consider yourself to be an expert on the topic you are blogging about before you write about it. If you don't know much about the topic, you might inadvertently share incorrect information, or information that is misleading. You have to also know enough about the topic to be able to respond in a smart way to those who comment on the blog post.Make sure you write about things people find interesting and useful. Ordinary chores like doing the laundry and Carry On Examining Through
This article explained how investing in the stock market can help you make substantial money. You'll be surprised of your earning when you finally get into the swing of investing. Take all of the tips you've learned here into consideration when making any investment decisions.Today's economy is failing. You can no longer rely on being able to get a job to be able to pay your bills. Most people today are lucky to make minimum wage, and anyone who makes more, tends to have so much debt that they can't survive. Included here are some tips on investing, which can help you to sustain your lifestyle. Avoid media programming that covers the stock market, from radio broadcasts to financial news networks. These outlets are great Carry On Examining Through
As with most technology, while games now give you more options they also can be quite complicated. Make sure that your television supports the HD capabilities that a lot of systems require. It might seem hard, but with a bit of research before buying anything, you will prevent problems. Do not slam your controllers or cause excess impact as this can cause damage. Controllers, like all electronic devices are very fragile. Do not throw your controller if you get angry since it might break as a result. Try different video games and stay up to date with new releases. A lot of players limit themselves to only on type of game and never try anything different. Trying out different video games helps you maximize your experience Carry On Examining Through
Understanding how gold is weighed can ensure that you don't get ripped off when selling your gold to a dealer. Gold can be weighed in Troy ounces, pennyweight, or grams. The most important thing is to make sure that whatever system the buyer is using to weigh your gold is the same one they're calculating the payment with.Never deal with a gold buyer or seller who does not have a favorable reputation. Their reputation can be verified though online sources, as well as the BBB. Do not do business with a buyer who does not treat you as a valued customer or does not offer a fair price. Protect your gold investment and yourself. If you are purchasing gold jewelry and have allergies to nickel or other metals, only purchase Carry On Examining Through
Cats love to play so make sure that you provide them with lots of toys. Cats enjoy batting around a fake mouse or little ball around the room. Some cats even develop a favorite toy. Playing with your cat is not only fun, it helps them to get exercise too.Dogs will wag their tails often to show their owners that they're excited to see them. While cats are also prone to tail wagging, they do it for very different reasons. Tail-wagging in a cat usually means a conflict or that they're deciding to flee or pounce regarding a situation. Along the same line, if your cat wags its tail while you are holding it, you should put it down to avoid getting scratched.Purchase high quality food for your cat. Your cat needs a lot of protein Carry On Examining Through
Practice your networking skills. MLM takes a lot of networking chops for real success. You've got to learn how to be personable, interesting, and yet still humble. And on top of it all, you've got to somehow sell without it feeling like a sale. This all takes practice to get it down.Make sure you have realistic earning expectations before you decide to get into multi-level marketing. Income generation and success is not as fruitful as many multi-level marketing companies make you believe. Statistically, only one out of 100 multi-level marketing representatives make any money or are successful. Furthermore, in the MLM world there are many shady business practices and scams, so be careful.Become educated about different Carry On Examining Through
If you don't track the statistics about who's watching your video, you can't possibly know what works and what doesn't. Monitor the effectiveness of each video you make, and compare them to each other as you go along. That will help you determine what works best and what just doesn't work for you; use the best working techniques going forward. Although the majority of your online videos are going to be short and should be that way, you should also provide longer, more in-depth videos. Keep the content of the short videos applicable to all your customers. A long video needs to really delve into a topic fully.Simply because you have keywords and transcripts in line with the key data in your video, you are not necessarily Carry On Examining Through
You can choose to hide your caller ID whenever you want to avoid showing it to someone that you call. First, access the Settings screen and select the Phone button. Next, tap the Show Caller ID button; it should now be disabled. That area of settings is also useful if you need to access your own phone number.The iPhone is popular thanks to everything it offers. The combination of features and apps make the iPhone fantastic for any busy person. The following tips will help you get more out of your iPhone. You may well feel that your iPhone's word suggestions cost you more time than they save. To end this feature, simply tap your iPhone screen to quit the suggestion box. What this means is that you aren't going to have Carry On Examining Through
When the manager of my apartment told me that my rent was going up again, I knew that I was going to have to move. Every year, the price keeps going up, and it was getting to the point where I could not afford it any longer. I started looking at other apartment complexes in the area, and it wasn't until I found http://treepoint-apts.com that I knew I had found the next place I wanted to live. The main reason is because so few apartments allow pets, and there was no way I was going to move without my dog. She means everything to me, and that is the only reason I didn't fight against the rent increases before. I thought it would be too hard to find another nice apartment that would allow me to bring her with me. I also Carry On Examining Through
Memory does tend to diminish as aging sets in, but there are certain techniques that can help one retain and improve memory function. Some fabulous brain fueling enhancers that reduce stress are well-balanced nutrition, adequate rest and relaxation, regular exercise, puzzles and strategy games and hearty laughter. You can find a number of useful books on the topic of memory improvement at your local public library. Many psychiatrists have written books that help people improve memory. You might not need anything more than these books to help.Pay a visit to the library in your area and pick up some books that will help you increase your memory. Many psychiatrists have written books that help people improve memory. You Carry On Examining Through