To find the fully factored form of the polynomial 32a^3 + 12a^2 + 4a^2 + 8a + 3 + 4a + 8a^2 + 3a + 12a^2 + 3a + 1 + 12a + 3a^2 + a, we first need to combine like terms and reorganize the expression.
Combining the like terms, we get:
- 32a^3
- (12a^2 + 4a^2 + 8a^2 + 12a^2 + 3a^2) = 39a^2
- (8a + 4a + 3a + 12a + a) = 28a
- + 3 (constant term)
Now the polynomial is:
32a^3 + 39a^2 + 28a + 3
Next, we can look for common factors or use polynomial long division or synthetic division to find roots. By testing small integer values (like 1 or -1) or using the Rational Root Theorem, we can find factors. After some trials, we may find that this polynomial can be decomposed into:
(8a + 3)(4a^2 + 5a + 1)
We can further check if the quadratic factor can be factored. The expression 4a^2 + 5a + 1 can be checked for roots and we find that it cannot be factored further with integers. Thus, our original polynomial can be expressed in its fully factored form as:
(8a + 3)(4a^2 + 5a + 1)
This gives us the simplest form presentable for an algebraic expression.