To factor the polynomial [tex]\(2x^3 + 16x^2 + 30x\)[/tex] completely, we will follow several steps methodically.
1. Identify the greatest common factor (GCF):
- The terms in the polynomial are [tex]\(2x^3\)[/tex], [tex]\(16x^2\)[/tex], and [tex]\(30x\)[/tex].
- The numerical coefficients are 2, 16, and 30. The GCF of these coefficients is 2.
- Each term also has an [tex]\(x\)[/tex], so the GCF of the variable part is [tex]\(x\)[/tex].
- Thus, the GCF of the entire polynomial is [tex]\(2x\)[/tex].
2. Factor out the GCF:
- We factor out [tex]\(2x\)[/tex] from each term in the polynomial:
[tex]\[
2x^3 + 16x^2 + 30x = 2x(x^2 + 8x + 15)
\][/tex]
3. Factor the quadratic [tex]\(x^2 + 8x + 15\)[/tex] inside the parentheses:
- To factor [tex]\(x^2 + 8x + 15\)[/tex], we need to find two numbers that multiply to 15 (the constant term) and add to 8 (the coefficient of the middle term [tex]\(x\)[/tex]).
- These two numbers are 3 and 5 because [tex]\(3 \times 5 = 15\)[/tex] and [tex]\(3 + 5 = 8\)[/tex].
- Therefore, the quadratic can be factored as:
[tex]\[
x^2 + 8x + 15 = (x + 3)(x + 5)
\][/tex]
4. Combine the factored parts:
- Putting it all together, we have:
[tex]\[
2x(x^2 + 8x + 15) = 2x(x + 3)(x + 5)
\][/tex]
Thus, the polynomial [tex]\(2x^3 + 16x^2 + 30x\)[/tex] is factored completely as [tex]\(2x(x + 3)(x + 5)\)[/tex].
The correct answer is:
(A) [tex]\(2x(x + 3)(x + 5)\)[/tex]
