Homework SourseFinish the proof of Theorem 430 if fx ax2 bx c epsilon Rx
Finish the proof of Theorem 4.30: if f(x) = ax^2 + bx + c epsilon R[x] with b^2 - 4ac ![Finish the proof of Theorem 4.30: if f(x) = ax^2 + bx + c epsilon R[x] with b^2 - 4ac SolutionWe know that as per the quadratic formula, the roots of f(x) are](/WebImages/41/finish-the-proof-of-theorem-430-if-fx-ax2-bx-c-epsilon-rx-1125379-1761599937-0.webp "Finish the proof of Theorem 4.30: if f(x) = ax^2 + bx + c epsilon R[x] with b^2 - 4ac SolutionWe know that as per the quadratic formula, the roots of f(x) are")
Solution
We know that as per the quadratic formula, the roots of f(x) are [ -b±(b2 -4ac)]/2a. Thus, if b2 -4ac < 0, then f(x) does not have real roots as -1 = i so that , if b2 -4ac = p (p < 0) , then the roots of f(x) are -b/2a ± i {(-p)}/2a which are both complex numbers. Thus f(x) is irreducible in R[x].
