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The stock market has dropped considerably over the past few months, and as a result, many stocks' dividend yields have become much more attractive. Our analysts feel that there are some strong long-term buying opportunities here in blue chips such as Conoco Phillips (COP -2.40%), ExxonMobil (XOM -2.88%), and Nucor (NUE -3.66%), and here's why.

Selena Maranjian: The price of oil recently plunged to below $43 per barrel, a six-year low, leading many people to continue writing off oil companies as possible investments. Don't be so hasty, though, as some, such as ConocoPhillips are worth considering. Oil prices won't stay low forever, and patient believers can collect some significant dividend income: ConocoPhillips recently yielded a whopping 6.5%.

The world's largest independent oil and gas producer is even well positioned if oil prices stay low for a long time, with an analyst from Argus suggesting, "Conoco's high-quality, low-cost asset base will help it to outperform peers when commodity prices are low." As the company aims to be cash-flow neutral by 2017, it's selling some non-core assets and bolstering its already-solid balance sheet. It's also cutting its costlier and more long-term projects such as its deepwater drilling, choosing to focus on higher-margin and shorter-term projects.

ConocoPhillips's last quarter was solid, thanks in part to significant cost-cutting. The company posted about $2 billion in operational free cash flow and its CEO Ryan Lance stressed in a conference call that, "The dividend is safe. Let me repeat that. The dividend is safe." Lance also directly addressed many people's worry, that oil prices will remain low for a long time:

We can exercise additional capital flexibility from various sources: deflation capture, efficiency improvements, discretion in development programs, uncommitted major projects, deepwater reductions and additional program efficiencies. And we believe we can achieve our 2017 production target giving the ramp from our major projects between now and then, the majority of which is from capital we've already invested. And this is all before tactical asset sales.

Tracing its roots back to 1875 and with a market cap recently topping $50 billion, ConocoPhillips is a big, dividend-paying blue-chip stock, and one worth considering.

Matt Frankel: Like Selena, I also have my eye on a big oil company, but I happen to prefer ExxonMobil. Oil may indeed stay cheap for several years, but not forever, and Exxon is a great way to set yourself up for success. I realize that ExxonMobil's 4.3% dividend yield pales in comparison with Conoco's, but there are other reasons to like Exxon.

First, there are some aspects of the oil business that actually benefit from lower oil prices, such as chemicals and refining -- two large parts of Exxon's diverse operations. And, the company is actually producing more oil. In its second quarter report, Exxon revealed that its overall production had increased by 3.6% year-over-year, and its liquids production increased by nearly 12%. Sure, this doesn't help now, but could set the company up to reap the benefits of an eventual rebound.

Finally, the reason I prefer Exxon is because it's in an excellent position to capitalize on the sector weakness through acquisitions. Consolidation is common when industrywide weakness strikes, and nobody is in a better position to go shopping than Exxon. The company has virtually unlimited access to funds with its AAA credit rating -- which is actually better than that of the U.S. government. In fact, Exxon raised $8 billion through a debt offering earlier this year, and has the power to raise more anytime it chooses.

Jason Hall: I suggest investors taking a closer look at steelmaker Nucor (NUE -3.66%).

Since last September, Nucor's stock is down 25%, as U.S. steelmakers face serious pressure from overseas competition. The domestic industry has for years now been crying foul over illegal dumping practices by importers, and the U.S. government has taken action, implementing tariffs on key steel imports in an effort to level the playing field.

Unfortunately, import levels haven't fallen, and actually increased in the first half of 2015. Nucor (and its industry cohorts) continue to contend that state-owned steelmakers in several countries are driving this oversupply, more focused on maintaining jobs at home, even if it means losing money by driving prices down.

So why is Nucor a blue chip to love in this messy market? Because it has shown, even in a very ugly downturn, that it can operate profitably and pay a consistent, regular dividend. The sell-off in the stock over the past year has only made its dividend yield even better, as it's now paying about 3.5% based on recent stock prices.

Don't get me wrong -- there's a chance that the stock could fall further in the short term, but that would be a product of market sentiment, not structural weakness in Nucor's business. For long-term investors looking for stable income, Nucor is a blue-chip stock that should be near the top of the list.