The roots of the quartic polynomial
are -2, 1, and -4.
To find the roots of the quartic polynomial
, we can use various methods such as factoring, synthetic division, or the use of the rational root theorem. In this case, let's use factoring to find the possible rational roots.
1. Rational Root Theorem:
The rational root theorem states that if a rational number p/q is a root of a polynomial, then p is a factor of the constant term (in this case, -8), and q is a factor of the leading coefficient (in this case, 1).
The factors of the constant term -8 are ±1, ±2, ±4, ±8.
The factors of the leading coefficient 1 are ±1.
2. Possible Rational Roots:
By testing the possible combinations of the factors obtained from the rational root theorem, we can find the possible rational roots of the polynomial. In this case, we need to test the following values:
±1/1, ±2/1, ±4/1, ±8/1
By substituting these values into the polynomial equation and simplifying, we can determine if they are roots or not.
3. Finding the Roots:
After testing all the possible rational roots, we find that the rational roots of the quartic polynomial
are:
x = -2, x = 1, and x = -4
These are the roots of the polynomial expressed in their simplest form.