To find the completely factored form of the expression 3x5 + 7x4 + 6x2 + 14x, we start by factoring out the greatest common factor (GCF) from all the terms.
1. **Identify the GCF**: The terms are:
- 3x5
- 7x4
- 6x2
- 14x
The coefficients are 3, 7, 6, and 14. The GCF of these coefficients is 1 (as they have no common factors other than 1). However, each term contains at least one factor of x. Therefore, we can factor out x from each term:
GCF = x
2. **Factoring out GCF**: Now, we can factor out x from the expression:
3x5 + 7x4 + 6x2 + 14x = x(3x4 + 7x3 + 6x + 14)
3. **Further factoring the polynomial**: Now the next step is to see if we can factor the polynomial inside the parentheses further:
The polynomial is 3x4 + 7x3 + 6x + 14.
To factor this polynomial, we can use techniques like synthetic division, grouping, or trial and error. In this case, it turns out that this polynomial does not factor neatly and remains in that form.
4. **Final result**: Therefore, the completely factored form of the original expression is:
x(3x4 + 7x3 + 6x + 14)
This is as far as we can go with factoring the given expression.