f(x)=x^4+6x^3+7x^2-6x-8 rational zeros, and factor af(x) into linear factors

Question

Given the polynomial function f(x) find the rational zeros, then the other zeros ( that is solve the equation f(x)=0) and factor into linear factors

Answer 1

Rational zeroes are ±1, ±2, ±4, ±8 (factors of 8). To find the actual zeroes we can use synthetic division.

Example:

1 | 1 6 7  -6 -8

     1 1 7 14  8

     1 7 14 8      so 1 is a zero (root).

Without going into details (for convenience), the same process can be used with all other potential zeroes. When this done, the following the zeroes emerge -4, -2, -1, 1 so f(x)=(x-1)(x+1)(x+2)(x+4).