Questions: Use synthetic division to divide x^3 + x^2 - x - 8 by x - 2

Solution Steps

To use synthetic division to divide \(x^3 + x^2 - x - 8\) by \(x - 2\), follow these steps:

1. Write down the coefficients of the polynomial: [1, 1, -1, -8].
2. Use the root of the divisor \(x - 2\), which is 2.
3. Perform synthetic division by bringing down the first coefficient, multiplying it by 2, adding the result to the next coefficient, and repeating the process.

We are dividing the polynomial \( P(x) = x^3 + x^2 - x - 8 \) by \( x - 2 \). The coefficients of the polynomial are \( [1, 1, -1, -8] \), and the root of the divisor is \( 2 \).

Using synthetic division, we start with the first coefficient \( 1 \) and proceed as follows:

1. Bring down the first coefficient: \( 1 \).
2. Multiply \( 1 \) by \( 2 \) (the root) and add to the next coefficient: \[ 1 \times 2 + 1 = 3 \]
3. Multiply \( 3 \) by \( 2 \) and add to the next coefficient: \[ 3 \times 2 - 1 = 5 \]
4. Multiply \( 5 \) by \( 2 \) and add to the last coefficient: \[ 5 \times 2 - 8 = 2 \]

The result of the synthetic division is \( [1, 3, 5, 2] \), where \( [1, 3, 5] \) are the coefficients of the quotient polynomial and \( 2 \) is the remainder.

The quotient polynomial can be expressed as: \[ Q(x) = 1x^2 + 3x + 5 \] The remainder is \( R = 2 \).

Thus, we can write the final result of the division as: \[ P(x) = (x - 2)(x^2 + 3x + 5) + 2 \]

Final Answer