Identify which of the following factors is a factor to x^3-x^2-52x+160 x+2 x-3 x+8 x+5 x-5 Now use synthetic division to find the missing factor and type it below. square

To identify the factors of the polynomial \(x^3 - x^2 - 52x + 160\), we can use synthetic division to test each given factor and see if it results in a remainder of zero.<br /><br />Let's start with \(x + 2\):<br /><br />1. Rewrite \(x + 2\) as \(x - (-2)\).<br />2. Set up synthetic division with \(-2\) as the divisor.<br /><br />\[<br />\begin{array}{r|rrrr}<br />-2 & 1 & -1 & -52 & 160 \\<br /> & & -2 & 6 & 92 \\<br />\hline<br /> & 1 & -3 & -46 & 252 \\<br />\end{array}<br />\]<br /><br />The remainder is 252, so \(x + 2\) is not a factor.<br /><br />Next, let's try \(x - 3\):<br /><br />1. Set up synthetic division with \(3\) as the divisor.<br /><br />\[<br />\begin{array}{r|rrrr}<br />3 & 1 & -1 & -52 & 160 \\<br /> & & 3 & 6 & -138 \\<br />\hline<br /> & 1 & 2 & -46 & 22 \\<br />\end{array}<br />\]<br /><br />The remainder is 22, so \(x - 3\) is not a factor.<br /><br />Next, let's try \(x + 8\):<br /><br />1. Rewrite \(x + 8\) as \(x - (-8)\).<br />2. Set up synthetic division with \(-8\) as the divisor.<br /><br />\[<br />\begin{array}{r|rrrr}<br />-8 & 1 & -1 & -52 & 160 \\<br /> & & -8 & 72 & -160 \\<br />\hline<br /> & 1 & -9 & 20 & 0 \\<br />\end{array}<br />\]<br /><br />The remainder is 0, so \(x + 8\) is a factor.<br /><br />Now that we have identified \(x + 8\) as a factor, we can use it to find the quotient polynomial. The quotient polynomial is \(x^2 - 9x + 20\).<br /><br />So, the missing factor is \(x + 8\).