Given the function P(x)=x^3+4x^2+kx+15 for what value of k is (x+3) a factor of P(x) -26 -8 8 26

To determine the value of \( k \) for which \((x+3)\) is a factor of \( P(x) = x^3 + 4x^2 + kx + 15 \), we can use the Factor Theorem. According to the Factor Theorem, \((x+3)\) is a factor of \( P(x) \) if and only if \( P(-3) = 0 \).<br /><br />Let's substitute \( x = -3 \) into the polynomial \( P(x) \) and set it equal to zero:<br /><br />\[ P(-3) = (-3)^3 + 4(-3)^2 + k(-3) + 15 \]<br /><br />Now, calculate each term:<br /><br />\[ (-3)^3 = -27 \]<br />\[ 4(-3)^2 = 4 \cdot 9 = 36 \]<br />\[ k(-3) = -3k \]<br /><br />So, the equation becomes:<br /><br />\[ -27 + 36 - 3k + 15 = 0 \]<br /><br />Combine the constants:<br /><br />\[ 24 - 3k = 0 \]<br /><br />Solve for \( k \):<br /><br />\[ -3k = -24 \]<br />\[ k = 8 \]<br /><br />Therefore, the value of \( k \) for which \((x+3)\) is a factor of \( P(x) \) is \( \boxed{8} \).