P(x) = 3x^3 - 7x^2 - 4x + 2
P(1 + √ 3) =
To evaluate P(1 + √3), we need to substitute the value 1 + √3 into the polynomial P(x) and simplify.
First, let's substitute 1 + √3 into the polynomial:
P(1 + √3) = 3(1 + √3)^3 - 7(1 + √3)^2 - 4(1 + √3) + 2
Now, we need to expand each term and simplify the expression.
(1 + √3)^3 can be expanded as:
(1 + √3)^3 = (1 + √3)(1 + √3)(1 + √3)
= (1 + √3)(1^2 + 2√3 + 3)
= (1 + √3)(4 + 2√3)
= 4 + 2√3 + 4√3 + 6
= 10 + 6√3
Similarly, (1 + √3)^2 can be expanded as:
(1 + √3)^2 = (1 + √3)(1 + √3)
= 1 + 2√3 + 3
= 4 + 2√3
Now, substituting the expanded values into the polynomial:
P(1 + √3) = 3(10 + 6√3) - 7(4 + 2√3) - 4(1 + √3) + 2
= 30 + 18√3 - 28 - 14√3 - 4 - 4√3 + 2
= -2 + (18√3 - 14√3 - 4√3) + (30 - 28)
= -2 + 0√3 + 2
= 0
Therefore, P(1 + √3) equals 0.