Final answer:
To solve the polynomial equation X⁴-6x³-5x²+24x+4=0, identify potential rational roots, confirm an actual root using synthetic division, and then factor or use the quadratic formula to find all roots of the equation.
Step-by-step explanation:
Given the polynomial equation X⁴-6x³-5x²+24x+4=0, we will follow these steps:
- List all possible rational roots using the Rational Root Theorem which states that any rational root, expressed in its lowest terms p/q, is such that p is a factor of the constant term (4) and q is a factor of the leading coefficient (1).
- Identify one actual root using synthetic division. This involves testing the possible rational roots until we find one that gives a result of zero.
- Upon finding an actual root, we can then use it to solve the equation by factoring or continuing with synthetic division to reduce the polynomial to a quadratic form, ax²+bx+c=0, and solve for the remaining roots.
Let's illustrate this process step by step:
- The possible rational roots are ± 1, ± 2, ± 4.
- Using synthetic division, we can test these possible roots. If synthetic division yields a remainder of zero, we have found an actual root.
- After identifying an actual root, let's say 'r', we can divide the polynomial by (x-r) to find the other roots. If the resulting polynomial is a quadratic equation of the form ax²+bx+c=0, the quadratic formula can be used to find the remaining roots.