For the expression \( 5 x^{3} y^{3}+x y^{2}+8 \) to be a trinomial with a degree of 5 , the missing exponent on the \( x \)-term must be

To make the expression \( 5 x^{3} y^{3}+x y^{2}+8 \) a trinomial with a degree of 5, we need to examine the existing terms. The first term, \( 5 x^{3} y^{3} \), has a total degree of \( 3 + 3 = 6 \), which is too high. The second term, \( x y^{2} \), has a degree of \( 1 + 2 = 3 \), and the constant \( 8 \) has a degree of 0. To raise the overall degree to 5, the second term \( x y^{2} \) must be adjusted. For \( x=\) term to have a degree sum of 5 when combined with \( y^{2} \), the exponent on \( x \) must be 3 since \( 3 + 2 = 5 \). Thus, we need to rewrite the second term as \( x^{3} y^{2} \). Therefore, the missing exponent on the \( x \)-term must be 3.