Question

Factor completely: x^3 + 4x^2 + 8x + 32.

A) (x + 4)(x^2 + 8)
B) (x - 4)(x^2 - 8)
C) (x - 4)(x^2 + 8)
D) (x + 4)(x^2 - 8)

Answer 1

The polynomial x^3 + 4x^2 + 8x + 32 can be factored completely by grouping, resulting in the factorization (x + 4)(x^2 + 8), which corresponds to option A.

Step-by-step explanation:

To factor the polynomial x^3 + 4x^2 + 8x + 32 completely, we look for common factors and patterns that resemble special products such as squares of binomials or difference of squares. By observing the coefficients (1, 4, 8, 32), we attempt to group them to find a common factor or use a method such as synthetic division or factoring by grouping.

First, we can attempt to factor by grouping:

x^3 + 4x^2 + 8x + 32
= x^2(x + 4) + 8(x + 4)
= (x^2 + 8)(x + 4)

Now we can see that the expression is factored completely and this corresponds to option A: (x + 4)(x^2 + 8). Therefore, the other options B, C, and D do not represent the complete factorization of the given polynomial. Remembering that the sum of squares is not factorable in the real number system, we can conclude that the expression cannot be further factored into real polynomial factors.