Question

Please Help! Determine the factors and not factors and roots and not roots of the following polynomial function.

f(x)= x^5-6x^4-16x^3+140x^2-249x+130

The answer bank is in the picture.
thank you.

Answer 1

Factors: (x - 2)(x - 5)(x - 1)(x - 3)(x + 1)

To determine the factors and roots of the given polynomial function f(x)=x^5 −6x^4 −16x^3 +140x^2 −249x+130, one can use the factor theorem and synthetic division. The factors and roots correspond to values of x that make the polynomial equal to zero.

The factor theorem states that if c is a root of the polynomial, then x−c is a factor. By testing potential roots using synthetic division, it is found that x=2, x=5, x=1, x=3, and x=−1 are roots of the polynomial, making (x−2), (x−5), (x−1), (x−3), and (x+1) factors, respectively.

Therefore, the factored form of the polynomial is (x−2)(x−5)(x−1)(x−3)(x+1). The roots of the polynomial are x=2, x=5, x=1, x=3, and x=−1. These values make the polynomial equal to zero, satisfying the definition of roots. The non-factors and non-roots correspond to values that do not result in zero when substituted into the polynomial.