Question 839106: Solve x^3-7x+4=0, please help thank you!
Answer by josh_jordan(263)   (Show Source):
You can put this solution on YOUR website! This is a polynomial that cannot be solved using the rational zero test and synthetic division. When you perform the rational zero test and synthetic division, you will realize that all of your factors of the constant, 4, produce no perfect result. So, in order to find your zeros, you will have to graph the function. After plugging in the polynomial into a graphing calculator, your graph will look like the following:
You will notice that there are 3 places in which the function crosses the x-axis: between -3 and -2, between 0 and 1, and between 2 and 3. Therefore, your zeroes will be decimals. Normally, when finding decimal zeroes, you should be accurate to 3 decimal places. So, that's what we will do. Let's look at the first place the function crosses the x-axis: between -3 and -2. It appears that it crosses somewhere closer to -3, between -2.5 and -3. So, let's find the number that is midway between -3 and -2.5. That would be -2.75. Let's plug in -2.75 in to our original equation, to determine the value: -2.75^3 - 7(-2.75) + 4 = 2.453125. Since our computed value is positive, we need to find a value between -3 and -2.75, since, based on our graph, the closer we get to -3 on the x-axis, our y-axis value gets closer to 0 from above the x-axis. So, let's find the number midway between -3 and -2.75. That would be -2.875. Now, repeat the process above by plugging in -2.875 into our equation. When we do this, our computed value is 0.361328125, which is alot closer to zero, but not close enough. So, we will need to try again, by plugging in a number into our equation that's midway between -3 and -2.875, which is -2.9375. Plugging that number into our equation gives us -0.78491210937, which is actually farther away than when we plugged in -2.875. This tells us that our zero must be between -2.875 and -2.9375. So, let's find the midway point between -2.875 and -2.9375, which is -2.90625. Plugging this into our original equation gives us -0.20327758789. We are getting so close! Remember, we need closer to zero, so we will find the midway point between -2.875 and -2.90625, which is -2.890625. Plugging this in to our original equation, yields 0.08114242553. That's even closer, but, let's do another midway point in between -2.90625 and -2.890625, since our last value equaled a positive number, which is -2.8984375. Plug this in to our original equation to give us -0.06053686141. We are so close, so let's try one more time. The point midway between -2.890625 and -2.8984375 is -2.89453125. Plugging that in to our original equation gives us 0.0104353. That's as close to 0 as we need to get. Note that even though that value is positive, the first three digits past the decimal point is only 1/100th off from being 0.000, so this is as close to 0 as we need to be. Therefore, rounding -2.89453125 to 3 decimal places, gives us -2.895
We are now going to repeat the process above for finding the other two zeros. When you repeat these steps, you will find your other two zeros, rounded to three decimal places, which are 0.603 and 2.292.
x ≈ -2.895, 0.603, and 2.292
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