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

Ask by Macdonald Summers. in the United States
Feb 27,2025

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Answer

The quotient is \( x^2 - 2x + 4 \) and the remainder is 0.

Solution

To perform synthetic division for (x³ - 4x² + 8x - 8) divided by (x - 2): 1. Write down the coefficients of the dividend: 1 (for x³), -4 (for x²), 8 (for x), and -8 (constant). 2. The divisor is (x - 2), so use 2 as the synthetic divisor. 3. Set up the synthetic division: 2 | 1 -4 8 -8 | 2 -4 8 ------------------------- 1 -2 4 0 Explanation: • Bring down the first coefficient (1). • Multiply 1 by 2 to get 2, then add this to -4 to obtain -2. • Multiply -2 by 2 to get -4, add to 8 to get 4. • Multiply 4 by 2 to get 8, add to -8 to obtain 0, which is the remainder. Thus, the quotient is x² - 2x + 4 and the remainder is 0.

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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: \( 0 \)

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