Question

Use either synthetic or long division to divide p(x) by d(x), and express p in the form p(x) = d(x) · q(x) + r(x). Given p(x) = 4x³ + 9x + 8 and d(x) = 2x + 1, find q(x) and r(x).

Answer 1

To divide p(x) by d(x) using synthetic division, arrange the terms, perform the division, and find the quotient and remainder.

Step-by-step explanation:

To divide p(x) = 4x³ + 9x + 8 by d(x) = 2x + 1 using synthetic division, follow these steps:

1. Arrange the terms of p(x) in descending order of the exponents.
2. Write down the coefficient of the highest power of x as the first row of synthetic division.
3. Write down the negation of the constant term of d(x) as the second row of synthetic division.
4. Perform the synthetic division by dividing the first term of the first row by the first term of the second row, and write the result on the third row.
5. Multiply the result from the third row by the second row, and write the product below the next term of the first row.
6. Add the numbers in the second and third rows, and write the sum on the fourth row.
7. Continue the synthetic division until all terms have been divided.
8. The quotient polynomial q(x) is formed by the coefficients in the fourth row, and the remainder r(x) is the last number in the fourth row.

By performing synthetic division, we find that p(x) = (4x² + x + 4) · (2x + 1) + (-4x + 4). Therefore, q(x) = 4x² + x + 4 and r(x) = -4x + 4.