Final answer:
The result of dividing the polynomial 2x^3 + 6x^2 + 6x + 2 by the binomial x + 1 is the polynomial 2x^2 + 4x + 2, obtained through the process of synthetic division.
Step-by-step explanation:
The question asks for the result of dividing the polynomial 2x^3 + 6x^2 + 6x + 2 by the binomial x + 1. To solve this, we can use either polynomial long division or synthetic division. Since the divisor is a binomial of the form x + a, synthetic division can be a quick method.
First, we set up the synthetic division by writing the coefficients of the polynomial: 2, 6, 6, and 2. Next, we use -1 for the division (the opposite sign of the 1 in x + 1).
The synthetic division process looks like this:
- Bring down the 2.
- Multiply -1 by 2 and write the result under the next coefficient, 6.
- Add this result to 6, write the sum, and repeat the multiplication and addition steps with the new numbers until you reach the end.
After completing the synthetic division, the result will be the coefficients of the quotient polynomial. In this case, the quotient will be 2x^2 + 4x + 2, and since the remainder is zero, there's no need to add any fraction to the quotient.
Remember the rules for multiplication and division of integers: two positive numbers or two negative numbers multiplied (or divided) together yield a positive result, while a positive and a negative number yield a negative result.