To find the other factor of the polynomial 3x² + 10x + 8, given that one factor is 3x + 4, we can use polynomial long division.
1. **Set up the division**: We will divide 3x² + 10x + 8 by 3x + 4.
2. **Divide the leading terms**: Take the leading term of the dividend, which is 3x², and divide it by the leading term of the divisor, which is 3x. This gives us x.
3. **Multiply and subtract**: Now multiply x by the entire divisor (3x + 4):
x(3x + 4) = 3x² + 4x.
Now, subtract this from the original polynomial:
(3x² + 10x + 8) – (3x² + 4x) = (10x – 4x) + 8 = 6x + 8.
4. **Repeat the process**: Now we need to divide the new leading term 6x by the leading term of the divisor 3x which gives us 2. Multiply 2 by the entire divisor:
2(3x + 4) = 6x + 8. Now subtract:
(6x + 8) – (6x + 8) = 0. There is no remainder.
5. **Conclusion**: The division gives us the result that the other factor is x + 2. Thus, we can express the polynomial 3x² + 10x + 8 as the product of its factors:
3x² + 10x + 8 = (3x + 4)(x + 2).
6. **Final Answer**: So, the other factor of the polynomial 3x² + 10x + 8, given that it has a factor of 3x + 4, is x + 2.