Let r1,r2,r3 be the roots of polynomial P(x)=x^3+3x+1. Evaluate the product
3
∏ (r^2+r +1)
k k
k=1
To evaluate the given product, we need to substitute the values of r1, r2, and r3 into the expression (r^2 + r + 1) and multiply them together for k = 1, 2, 3.
First, let's find the values of r1, r2, and r3 by solving the polynomial equation P(x) = 0.
P(x) = x^3 + 3x + 1
To find the roots, we can use numerical methods or use a calculator. For this explanation, let's use a calculator.
By solving the equation P(x) = 0, we find that the roots are approximately:
r1 ≈ -1.532, r2 ≈ 0.266, r3 ≈ 1.266.
Now, let's substitute these values into the expression (r^2 + r + 1) and multiply them together for k = 1, 2, 3:
For k = 1:
Substituting r1 = -1.532 into (r^2 + r + 1):
(-1.532)^2 + (-1.532) + 1 = 1.333
For k = 2:
Substituting r2 = 0.266 into (r^2 + r + 1):
(0.266)^2 + (0.266) + 1 = 1.232
For k = 3:
Substituting r3 = 1.266 into (r^2 + r + 1):
(1.266)^2 + (1.266) + 1 = 4.531
Finally, we evaluate the product:
∏(r^2 + r + 1) = (1.333) * (1.232) * (4.531) ≈ 8.82