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

To identify which of the given factors is a factor of the polynomial \(x^3 - x^2 - 52x + 160\), we can use the Factor Theorem. The Factor Theorem states that if \(x = a\) is a root of the polynomial \(P(x)\), then \((x - a)\) is a factor of \(P(x)\).<br /><br />Let's test each given factor by substituting the corresponding root into the polynomial and checking if it equals zero.<br /><br />1. For \(x + 2\), the root is \(x = -2\):<br /> \[<br /> P(-2) = (-2)^3 - (-2)^2 - 52(-2) + 160 = -8 - 4 + 104 + 160 = 252 \neq 0<br /> \]<br /> So, \(x + 2\) is not a factor.<br /><br />2. For \(x - 3\), the root is \(x = 3\):<br /> \[<br /> P(3) = 3^3 - 3^2 - 52(3) + 160 = 27 - 9 - 156 + 160 = 22 \neq 0<br /> \]<br /> So, \(x - 3\) is not a factor.<br /><br />3. For \(x + 8\), the root is \(x = -8\):<br /> \[<br /> P(-8) = (-8)^3 - (-8)^2 - 52(-8) + 160 = -512 - 64 + 416 + 160 = 0<br /> \]<br /> So, \(x + 8\) is a factor.<br /><br />4. For \(x + 5\), the root is \(x = -5\):<br /> \[<br /> P(-5) = (-5)^3 - (-5)^2 - 52(-5) + 160 = -125 - 25 + 260 + 160 = 270 \neq 0<br /> \]<br /> So, \(x + 5\) is not a factor.<br /><br />5. For \(x - 5\), the root is \(x = 5\):<br /> \[<br /> P(5) = 5^3 - 5^2 - 52(5) + 160 = 125 - 25 - 260 + 160 = 0<br /> \]<br /> So, \(x - 5\) is a factor.<br /><br />Therefore, the factors of the polynomial \(x^3 - x^2 - 52x + 160\) are \(x + 8\) and \(x - 5\).