Use the rational zeros theorem to find all the real zeros of

Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x) = x^3 - x^2 - 37x - 35 Find the real zeros of f. Select the correct choice below and; if necessary, fill in the answer box to complete your answer. (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression. Use a comma to separate answers as needed.) There are no real zeros. Use the real zeros to factor f. f(x)= (Simplify your answer. Type your answer in factored form. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression.)

Solution

f(x) = x^3 - x^2 -37x -35

use rational root theorem :

The factor of the leading coefficient (1) is 1 .The factors of the constant term (-35) are 1 5 7 35 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±5/1, ±7/1, ±35/1

check for all values of x in f(x):

f(-1) =0

Divide f(x)/(x+1) = (x^3 - x^2 -37x -35)/(x+1)

= x^2 -2x -35

solve the quadratic: we get x= -5 , 7

So, rational roots of f(x) ; x=-1 , -5, 7