Final answer:
After applying synthetic division to the polynomial x^3+12x^2+47x+60 using the root -5, associated with the factor (x+5), the result is x^2 + 7x + 12. This is the other factor representing the given polynomial, corresponding to option A.
Step-by-step explanation:
Since (x+5) is a factor of x^3+12x^2+47x+60, we can find the other factor by performing polynomial division or applying synthetic division. Here's the process using synthetic division:
- Write down the coefficients of the polynomial: 1 (for x^3), 12 (for x^2), 47 (for x), and 60 (constant term).
- Since x+5 is a factor, the root associated with this factor is -5.
- Apply synthetic division using the root -5 and the coefficients of the polynomial.
The remainder after the synthetic division, in this case, should be zero. The result of the synthetic division gives us the coefficients of the quadratic polynomial which forms the other factor of the given polynomial. Following this process:
- Bring down the leading coefficient (1).
- Multiply -5 by 1 and add the result to the next coefficient (12), giving us -5 + 12 = 7.
- Multiply -5 by 7 and add the result to the next coefficient (47), giving us -35 + 47 = 12.
- Lastly, multiply -5 by 12 and add the result to the final coefficient (60), which should be net zero since x+5 is a factor.
Therefore, the quotient of the synthetic division and hence the other factor of the polynomial is x^2 + 7x + 12, which corresponds to option A.