Question

Determine the number and type of roots for the equation using one of the given roots. Then find each root. x^3−7x+6=0 ; x=1

a) One real root
b) Two real roots and one imaginary root
c) Three real roots
d) Two imaginary roots

Answer 1

The given equation x^3−7x+6=0 can be factored as (x-1)(x^2+x-6)=0, leading to the three roots: x=1, x=-3, and x=2.

Step-by-step explanation:

The given equation is x^3−7x+6=0. To determine the number and type of roots, we can use the rational root theorem, which states that if a rational number p/q is a root of the equation, then p must be a factor of the constant term (in this case, 6) and q must be a factor of the leading coefficient (in this case, 1).

By trying out different rational numbers as potential roots, we can find that x=1 is a root of the equation. By performing polynomial division, we can factor the equation as (x-1)(x^2+x-6)=0.

Setting each factor equal to zero, we can solve for x and find the three roots of the equation: x=1, x=-3, and x=2.