Respuesta :
Answer:
Step-by-step explanation:
To find the factors of the polynomial P(x) = x^4 - 5x^3 + 3x + 2, we can start by checking for possible rational roots using the Rational Root Theorem. The Rational Root Theorem states that any rational root of a polynomial equation is in the form of p/q, where p is a factor of the constant term (in this case, 2) and q is a factor of the leading coefficient (in this case, 1).
1. Factors of the constant term (2): ±1, ±2
2. Factors of the leading coefficient (1): ±1
Combining these factors gives us the possible rational roots: ±1, ±2
To test these possible roots, we can use synthetic division or polynomial long division. We substitute each possible root into the polynomial P(x) and check if the result is equal to zero. If it is, then that value is a root of the polynomial.
After testing the possible roots, the actual roots of the polynomial can be found by factoring out the root we found using synthetic division or long division. This process can be repeated to factorize the polynomial completely.
By factoring the polynomial completely, we can express it as the product of linear factors and possibly irreducible quadratic factors.
In this case, if a rational root is found and factored out, the polynomial P(x) can be expressed as a product of linear and quadratic factors. This process helps us understand the structure of the polynomial and find its roots.
