Final answer:
The polynomial (x³ - 13x² + 40x + 18) can be divided by (x - 7) using synthetic division to yield the quotient (x² - 6x - 2), making the correct answer A.
Step-by-step explanation:
The question is asking for the division of a polynomial by a binomial, which is a basic operation in algebra. The polynomial (x³ - 13x² + 40x + 18) needs to be divided by the binomial (x - 7). To find the answer, one can either use polynomial long division or synthetic division. Both methods will yield the quotient of the division.
To demonstrate, we'll use synthetic division:
- Set up the synthetic division by writing down the coefficients of the polynomial, which are 1, -13, 40, and 18, and the root of the binomial which is 7.
- Bring down the first coefficient, which is 1.
- Multiply the root by the new quotient's coefficient, and write the result under the next coefficient. For the first step, we multiply 1 by 7 to get 7, and write it under -13.
- Add the numbers in the second column to get the new coefficient, which is -13 + 7 = -6.
- Continue this process for all coefficients. Multiply 7 by the new quotient's coefficient, -6, to get -42; write this under 40, then add to get -2. Next, multiply -2 by 7 to get -14 and write this under 18, then add to get 4.
- As a result, the quotient is (x² - 6x - 2), with a remainder of 4.
The correct answer is A. (x² - 6x - 2) plus the remainder should be written as (x² - 6x - 2) + 4/(x - 7).