Find the rational roots of x4 + 8x3 + 7x2 – 40x – 60 = 0.
To find the rational roots, we can use the Rational Root Theorem. According to the theorem, if the rational root of the polynomial equation x4 + 8x3 + 7x2 – 40x – 60 = 0 exists, it must be of the form p/q, where p is a factor of the constant term -60 and q is a factor of the leading coefficient 1.
The factors of -60 are ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, and ±60.
The factors of 1 are ±1.
So, the possible rational roots are:
±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, and ±60.
We can now test each of these values by substituting them into the equation x4 + 8x3 + 7x2 – 40x – 60 = 0 and checking for which values the equation equals zero.
After testing the values, we find that the rational roots of the equation x4 + 8x3 + 7x2 – 40x – 60 = 0 are:
x = -5 and x = -2.