(P) 3 The coefficient of \( x^{3} \) in the expansion of \( (3+b x)^{5} \) is -720 . Find the value of the constant \( b \). (P) 4 The coefficient of \( x^{3} \) in the expansion of \( (2+x)(3-a x)^{4} \) is 30 . Find the three possible values of the constant \( a \). (E/P) 5 When \( (1-2 x)^{p} \) is expanded, the coefficient of \( x^{2} \) is 40 . Given that \( p>0 \), use this information

To solve the problem of finding \( b \) in the expression \( (3 + b x)^{5} \), we can use the binomial expansion formula. The general term in this expansion is given by \( T_k = \binom{n}{k} a^{n-k} b^k \), where \( n \) is the exponent, and \( a \) and \( b \) are the two parts of the binomial. Here, we substitute \( n = 5 \), \( a = 3 \), and \( k = 3 \) to find the coefficient of \( x^3 \). The expression simplifies to \( \binom{5}{3} \cdot 3^{2} \cdot (b)^{3} \). Setting this equal to -720 allows us to solve for \( b \). Exploring the \( (2+x)(3-a x)^{4} \) expansion brings another layer of complexity as we apply the product rule of binomials. We first expand \( (3 - ax)^{4} \) and identify the terms generating \( x^3 \). The coefficients of relevant terms from both products combine to yield an equation equal to 30, leading us to solve a cubic equation for \( a \). Keep your algebra skills sharp and watch out for signs when calculating those coefficients!