If r sub1, r sub2, r sub3, r sub4...?

Question

If r sub1, r sub2, r sub3, r sub4 are roots of x^4 - 4x^2 + 2 = 0, what is the value of (1+r sub1)(1+r sub2)(1+r sub3)(1+r sub4)?

Answer 1

Let's consider the function:

f(x) = (x + r₁)(x + r₂)(x + r₃)(x + r₄)

And notice that we must calculate f(1).

Since r₁, r₂, r₃ and r₄ are roots of g(x) = x⁴ - 4x² + 2, that necessarily means that f(x) = g(x) for all x because f(x) is g(x) in factored form.

Hence,

f(1) = g(1) = (1)⁴ - 4(1)² + 2 = 1 - 4 + 2 = -1

Therefore,

(1 + r₁)(1 + r₂)(1 + r₃)(1 + r₄) = -1

Hope this helps!