To find the factor form of the polynomial \(f(x) = x^3 - x^2 - 37x - 35\), you can start by looking for its rational roots using the Rational Root Theorem. The Rational Root Theorem states that any rational root of the polynomial must be a factor of the constant term (in this case, -35) divided by a factor of the leading coefficient (in this case, 1).
The factors of 35 are ±1, ±5, ±7, and ±35, and the factors of 1 are ±1. So, the possible rational roots are:
\(\pm1, \pm5, \pm7, \pm35\)
Now, you can use synthetic division or long division to test these possible roots and see which ones, if any, are roots of the polynomial.
After performing the division, you'll find that \(x = -5\) is a root of the polynomial. So, \(x + 5\) is one of the factors.
Now, you can perform polynomial division to factor \(f(x)\) further:
\(f(x) = x^3 - x^2 - 37x - 35 = (x + 5)(x^2 - 6x - 7)\)
Next, you can factor the quadratic \(x^2 - 6x - 7\) using the quadratic formula or by factoring:
\(x^2 - 6x - 7 = (x - 7)(x + 1)\)
So, the factor form of \(f(x)\) is:
\(f(x) = (x + 5)(x - 7)(x + 1)\)