Using synthetic division , solve and select the remainder. (x^3-x^2-5x+1)div (x+1) A 0 B -5 C 4 (C) 4 D -2

To solve the problem using synthetic division, we need to divide the polynomial \(x^3 - x^2 - 5x + 1\) by \(x + 1\). In synthetic division, we use the root of the divisor, which is \(-1\) in this case (since \(x + 1 = 0\) gives \(x = -1\)).<br /><br />Here are the steps for synthetic division:<br /><br />1. Write down the coefficients of the dividend: \(1, -1, -5, 1\).<br />2. Use \(-1\) as the divisor.<br />3. Bring down the leading coefficient (1) to the bottom row.<br />4. Multiply \(-1\) by the number just written on the bottom row and write the result under the next coefficient.<br />5. Add the numbers in the second column and write the result in the bottom row.<br />6. Repeat the process for each column.<br /><br />Let's perform the synthetic division:<br /><br />\[<br />\begin{array}{r|rrrr}<br />-1 & 1 & -1 & -5 & 1 \\<br /> & & -1 & 2 & 3 \\<br />\hline<br /> & 1 & -2 & -3 & 4 \\<br />\end{array}<br />\]<br /><br />The bottom row represents the coefficients of the quotient polynomial, and the last number is the remainder. Therefore, the remainder is 4.<br /><br />Thus, the correct answer is:<br />\[<br />\boxed{4}<br />\]