Final answer:
The result of dividing 3x^3 - 19x^2 - 12x - 14 by x - 7 using synthetic division is 3x^2 - 40x + 278. This corresponds to option (b) in the question. The provided reference information is irrelevant to the division process.
Step-by-step explanation:
To determine the result when dividing the polynomial 3x3 - 19x2 - 12x - 14 by x - 7, we can use either polynomial long division or synthetic division. Since the divisor is of the form x - a, where a is a constant, synthetic division is a more efficient method.
In synthetic division, we set up the process by writing the coefficients of the polynomial in descending order of their powers. We then bring down the leading coefficient and begin the synthetic division process by multiplying and adding as per the synthetic division rules. Unfortunately, the provided reference information is for solving a quadratic equation using the quadratic formula, which is not applicable to the given polynomial division problem. So, we will disregard that information and proceed with synthetic division.
The steps involved in synthetic division are:
- Write down the coefficients of the polynomial: 3, -19, -12, -14.
- Place the zero of the divisor x - 7, which is 7, to the left.
- Bring down the first coefficient, which is 3.
- Multiply 7 by the number just written below the line (initially 3) and write the result below the next coefficient.
- Add the numbers in each column and write the result below the line.
- Repeat this process for the remaining coefficients.
- The numbers you get as a result are the coefficients of the quotient polynomial.
After performing synthetic division, the quotient is found to be 3x2 - 40x + 278, which corresponds to option (b).