Find the quotient and remainder using synthetic division. \[ \frac{x^{3}-4 x^{2}+8 x-8}{x-2} \] quotient remainder

To perform synthetic division of \( x^{3}-4 x^{2}+8 x-8 \) by \( x-2 \), we will use 2 (the root of \( x-2=0 \)) and the coefficients of the polynomial: 1 (for \( x^3 \)), -4 (for \( x^2 \)), 8 (for \( x \)), and -8 (constant term). 1. Write down the coefficients: \( 1, -4, 8, -8 \). 2. Bring down the leading coefficient (1). 3. Multiply 2 (the divisor) by 1 and write the result under the next coefficient: \( 2 \times 1 = 2 \). 4. Add: \( -4 + 2 = -2 \). 5. Multiply 2 by -2: \( 2 \times -2 = -4 \). 6. Add: \( 8 + (-4) = 4 \). 7. Multiply 2 by 4: \( 2 \times 4 = 8 \). 8. Add: \( -8 + 8 = 0 \). There are no more coefficients to bring down, so the process stops here. The row of results is \( 1, -2, 4, 0 \). Thus, the quotient is \( x^{2} - 2x + 4 \) and the remainder is \( 0 \). Quotient: \( x^{2} - 2x + 4 \) Remainder: \( 0 \)