Use synthetic division to find the quotient and the remainder. \( \left(x^{4}-16\right) \div(x-2) \)

To use synthetic division to divide \( x^4 - 16 \) by \( x - 2 \), we start by setting up synthetic division using the root of the divisor, which is \( x = 2 \). The coefficients of \( x^4 - 16 \) can be expressed as \( 1, 0, 0, 0, -16 \) (noting that there are missing coefficients for \( x^3, x^2, \) and \( x^{1} \)). Setting up synthetic division: ``` 2 | 1 0 0 0 -16 | 2 4 8 16 --------------------- 1 2 4 8 0 ``` 1. Bring down the 1, multiply by 2 to get 2, add to the next coefficient (0), resulting in 2. 2. Multiply 2 by 2 to get 4, add to the next coefficient (0), resulting in 4. 3. Multiply 4 by 2 to get 8, add to the next coefficient (0), resulting in 8. 4. Multiply 8 by 2 to get 16, add to the last coefficient (-16), resulting in 0. The final line gives us the coefficients of the quotient: \( 1, 2, 4, 8 \). Thus, the quotient is \( x^3 + 2x^2 + 4x + 8 \) and the remainder is \( 0 \). **Quotient:** \( x^3 + 2x^2 + 4x + 8 \) **Remainder:** \( 0 \)