To find the root of the equation X^3 + X^2 + X + 7 = 0, we first need to identify an interval [a, b] that contains the root by checking the sign change of the function at the end points. Let's test for x = -2 and x = 0:
f(-2) = (-2)^3 + (-2)^2 + (-2) + 7 = -8 + 4 - 2 + 7 = 1
f(0) = 0^3 + 0^2 + 0 + 7 = 7
Since f(-2) > 0 and f(0) > 0, we know that the root lies in the interval [-2, 0].
Now, we can start the bisection method to find the root:
1. Calculate the midpoint of the interval: c = (a + b) / 2 = (-2 + 0) / 2 = -1
2. Calculate f(c): f(-1) = (-1)^3 + (-1)^2 + (-1) + 7 = -1 + 1 - 1 + 7 = 6
Since f(-1) > 0, the new interval becomes [a, -1].
3. Calculate the new midpoint: c = (-2 - 1) / 2 = -1.5
4. Calculate f(c): f(-1.5) = (-1.5)^3 + (-1.5)^2 + (-1.5) + 7 = -3.375 + 2.25 - 1.5 + 7 = 4.375
Since f(-1.5) > 0, the new interval becomes [-2, -1.5].
Repeat these steps until the root is found within the desired accuracy.
After several iterations, we find that the root of the equation X^3 + X^2 + X + 7 = 0 correct to three decimal places is approximately x = -1.843.
Obtain the root correct to three decimal places for each of the following equations using bisection method X3+x²+x+7=0
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