Consider the polynomial p(x)=8x3+12x2+2x+3.
Part A: What is the correct factorization of p(x)=8x3+12x2+2x+3 over the integers?
Part B: What method is used to factor p(x)=8x3+12x2+2x+3?
Select one answer for Part A and select one answer for Part B.
B: perfect-square trinomial
B: grouping
A: (2x+3)(4x+1)
B: sum of cubes
A: (4x2+1)(2x+3)
A: (2x+3)(4x2+6x+1)
Part A: The correct factorization of p(x) = 8x^3 + 12x^2 + 2x + 3 over the integers is (2x + 3)(4x^2 + 1).
To arrive at this factorization, we look for common factors by checking if any of the integers divide each term evenly. In this case, there are no common factors other than 1.
Next, we can try to factor by grouping, which is not applicable in this case.
Finally, we can use the method of recognizing a difference of squares. However, there are no perfect squares in this polynomial.
Hence, the factorization of p(x) = 8x^3 + 12x^2 + 2x + 3 over the integers is (2x + 3)(4x^2 + 1).
Part B: The method used to factor p(x) = 8x^3 + 12x^2 + 2x + 3 is not a perfect-square trinomial, sum of cubes, or grouping. The factorization in Part A does not correspond to any of these methods.
Instead, this particular polynomial can be factored using the method of recognizing a common factor and then applying a factoring technique called "factoring by grouping." However, in this case, there are no common factors other than 1, and factoring by grouping is not applicable.
Hence, the correct answer for Part B is none of the given options. The polynomial p(x) = 8x^3 + 12x^2 + 2x + 3 is not factored using any of the listed methods.