3 What are all the roots of \( x^{3}-3 x^{2}-x+3=0 \) Whaice 1 point

To find the roots of the polynomial \( x^{3}-3 x^{2}-x+3=0 \), we can use the Rational Root Theorem to test possible rational roots. By substituting \( x = 1 \), we find that \( 1 \) is indeed a root. Factoring \( x-1 \) out of the polynomial, we can perform polynomial long division and find the remaining quadratic \( x^{2}-2x-3 \). We can further factor this as \( (x-3)(x+1) \). Therefore, the complete set of roots is \( x = 1, 3, -1 \). Roots of polynomials can paint a picture of various phenomena in the real world; for instance, they can represent equilibrium points in economics, such as supply and demand equilibrium, or predict behavior in engineering systems, like the stability of structures. Understanding polynomial behavior at these roots helps in optimizing solutions in various applications. If you're solving polynomials like a pro, remember that the sign changes in synthetic division can lead to common mistakes, such as miscalculating remainders or not recognizing a factor. Always double-check your synthetic division steps, factor confirmations, and use the discriminant to find out if the quadratic can be easily solved!