Question

Use the Remainder Theorem to completely factor p(x) = x 3 + 7x 2 + 11x + 5.

Answer 1

Let's try (x+5) first. According to the Remainder Theorem, if (x+5) is a factor of the polynomial, the remainder is zero. 

when x+5=0, x=-5
use synthetic division:

-5 I  1    7    11     5
    I        -5    -10   -5
     ----------------------
        1    2     1      0
the remainder is 0, so (x+5) is a factor:
the result is (x+5)(x²+2x+1)
factor again: (x+5)(x+1)(x+1) is the final answer.