The polynomial p(x)=-2x^3+3x+4 is given. Which conclusion is valid if p(1)=5 (x-1) is a factor of p(x) (x-5) is a factor of p(x) The remainder of p(x) divided by (x-1) is 5. The remainder of p(x) divided by (x-5) is 1.

To determine which conclusion is valid, we need to evaluate the polynomial \( p(x) = -2x^3 + 3x + 4 \) at the given points and check the remainders.<br /><br />First, let's evaluate \( p(1) \):<br />\[ p(1) = -2(1)^3 + 3(1) + 4 = -2 + 3 + 4 = 5 \]<br /><br />Since \( p(1) = 5 \), according to the Remainder Theorem, the remainder of \( p(x) \) divided by \( (x-1) \) is 5. Therefore, the statement "The remainder of \( p(x) \) divided by \( (x-1) \) is 5" is valid.<br /><br />Next, let's check if \( (x-1) \) is a factor of \( p(x) \). For \( (x-1) \) to be a factor, \( p(1) \) must be 0, but we found that \( p(1) = 5 \). Therefore, \( (x-1) \) is not a factor of \( p(x) \).<br /><br />Now, let's evaluate \( p(5) \):<br />\[ p(5) = -2(5)^3 + 3(5) + 4 = -2(125) + 15 + 4 = -250 + 15 + 4 = -231 \]<br /><br />Since \( p(5) = -231 \), the remainder of \( p(x) \) divided by \( (x-5) \) is -231, not 1. Therefore, the statement "The remainder of \( p(x) \) divided by \( (x-5) \) is 1" is not valid.<br /><br />In summary, the valid conclusion is:<br />- The remainder of \( p(x) \) divided by \( (x-1) \) is 5.