That only works if all exponents are even. Since you have an x³ term in there, then that won't help you.

I looked at the "exact" solution, and it's a mess. So the way you are probably meant to solve this is to use Newton's method.

That method has you start with a guess (x₀), then feed the guess through this equation:

x₁ = x₀ - f(x₀) / f'(x₀)

To get a better guess. Then use that as an input into the same equation to get a better guess, etc. until you have an answer as precise as you need.

So since we have this as our equation:

f(x) = x⁴ - x³ - x² - x - 1

We then need the first derivative, which is:

f'(x) = 4x³ - 3x² - 2x - 1

So substituting these expressions in for the functions, we get:

x₁ = x₀ - (x₀⁴ - x₀³ - x₀² - x₀ - 1) / (4x₀³ - 3x₀² - 2x₀ - 1)

Looking at the graph, we have two roots, one near -0.75 and the other near 1.9

I'll do the first one, you can follow the same steps with the other as the starting guess to get the second root. So I'll use x₀ = 1.9 to solve for x₁:

x₁ = 1.9 - (1.9⁴ - 1.9³ - 1.9² - 1.9 - 1) / (4 * 1.9³ - 3 * 1.9² - 2 * 1.9 - 1)

x₁ = 1.9 - (13.0321 - 6.859 - 3.61 - 1.9 - 1) / (4 * 6.859 - 3 * 3.61 - 2 * 1.9 - 1)

x₁ = 1.9 - (-0.3369) / (27.436 - 10.83 - 3.8 - 1)

x₁ = 1.9 - (-0.3369) / 11.806

x₁ = 1.9 + 0.028536

x₁ = 1.928536

You'll notice that a small value was added to our first guess. As we loop through, what gets added or subtracted from our previous guess will get smaller and smaller so you know you are getting closer and closer to the exact solution.

I'll go through two more times:

x₂ = x₁ - (x₁⁴ - x₁³ - x₁² - x₁ - 1) / (4x₁³ - 3x₁² - 2x₁ - 1)

x₂ = 1.928536 - (1.928536⁴ - 1.928536³ - 1.928536² - 1.928536 - 1) / (4 * 1.928536³ - 3 * 1.928536² - 2 * 1.928536 - 1)

x₂ = 1.928536 - (13.832829 - 7.17271 - 3.71925 - 1.928536 - 1) / (4 * 7.17271 - 3 * 3.71825 - 2 * 1.928536 - 1)

x₂ = 1.928536 - 0.012333 / (28.69084 - 11.15475 - 3.857072 - 1)

x₂ = 1.928536 - 0.012333 / 12.679018

x₂ = 1.928536 - 0.000972709

x₂ = 1.927563

Since this was 9 times 10^(-4), we should be accurate to 3SF now. Let's go one more and that will get us more accurate:

x₃ = x₂ - (x₂⁴ - x₂³ - x₂² - x₂ - 1) / (4x₂³ - 3x₂² - 2x₂ - 1)

x₃ = 1.927563 - (1.927563⁴ - 1.927563³ - 1.927563² - 1.927563 - 1) / (4 * 1.927563³ - 3 * 1.927563² - 2 * 1.927563 - 1)

x₃ = 1.927563 - (13.804934 - 7.161858 - 3.715499 - 1.927563 - 1) / (4 * 7.161858 - 3 * 3.715499 - 2 * 1.927563 - 1)

x₃ = 1.927563 - 0.000014 / (28.647432 - 11.146497 - 3.855126 - 1)

x₃ = 1.927563 - 0.000014 / 12.645809

x₃ = 1.927563 - 0.00000110709

x₃ = 1.927562

So after rounding to 6DP, we only changed one digit from the last guess. So we know this is good to 6SF.

So if you do the same process with the same equation using the starting guess of -0.75 and go through it three times, you should get a value for the other root.

If this helped, please give best answer. Thanks.