-1 is a zero of a polynomial p(x) p(x)=x^6+kx^4-x^3+3x Find k. square

To find the value of \( k \) such that \(-1\) is a zero of the polynomial \( p(x) = x^6 + kx^4 - x^3 + 3x \), we need to substitute \( x = -1 \) into the polynomial and set it equal to zero.<br /><br />Given:<br />\[ p(x) = x^6 + kx^4 - x^3 + 3x \]<br /><br />Substitute \( x = -1 \):<br />\[ p(-1) = (-1)^6 + k(-1)^4 - (-1)^3 + 3(-1) \]<br /><br />Calculate each term:<br />\[ (-1)^6 = 1 \]<br />\[ k(-1)^4 = k \cdot 1 = k \]<br />\[ (-1)^3 = -1 \]<br />\[ 3(-1) = -3 \]<br /><br />Now, substitute these values back into the equation:<br />\[ p(-1) = 1 + k - (-1) - 3 \]<br />\[ p(-1) = 1 + k + 1 - 3 \]<br />\[ p(-1) = k - 1 \]<br /><br />Since \(-1\) is a zero of the polynomial, we have:<br />\[ p(-1) = 0 \]<br /><br />Therefore:<br />\[ k - 1 = 0 \]<br />\[ k = 1 \]<br /><br />So, the value of \( k \) is:<br />\[ \boxed{1} \]