Final answer:
Use synthetic division by substituting -2 for x in the polynomial 4x^3 + 3x^2 - 19x + 18 to find the result when divided by x + 2. The process involves multiplying and adding coefficients to get the result, which will be a quadratic polynomial if there is no remainder.
Step-by-step explanation:
Synthetic Division of a Polynomial:
To use synthetic division to divide the polynomial 4x^3 + 3x^2 - 19x + 18 by the binomial x + 2, we first need to set up the synthetic division. The binomial 'divisor' is x + 2, which implies our synthetic substitution value is -2 since it would make the binomial equal to zero. We then write the coefficients of the polynomial in a row: 4, 3, -19, 18. performing the synthetic division, we bring down the first coefficient, 4, and then follow these steps:
- Multiply the value brought down by -2 (the value that x + 2 is set to zero by) and write the result under the next coefficient.
- Add the second column numbers together and write the sum under the same column.
- Repeat this process for each column.
The final row will contain the coefficients of the quotient polynomial.
Division Results:
After performing the synthetic division, the result will be a polynomial of one degree less than the original polynomial, which in this case is a quadratic polynomial, and possibly a remainder if the original polynomial is not evenly divisible by the binomial.