Use the binomial theorem to expand the following. $(6 x+7)^{4}
Answer
Finally, we check our result. The degree of the polynomial is 4, which is the same as the exponent in the original expression, so our result meets the requirements of the problem.
Step 1 ：First, we need to understand the binomial theorem. The binomial theorem states that for any positive integer \(n\), the expansion of \( (a+b)^n \) is given by \( (a+b)^n = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + \ldots + b^n \).
Step 2 ：In this problem, \(a = 6x\) and \(b = 7\), and \(n = 4\).
Step 3 ：We can substitute these values into the binomial theorem to get the expansion of \( (6x+7)^4 \).
Step 4 ：The first term is \( (6x)^4 = 1296x^4 \).
Step 5 ：The second term is \( \binom{4}{1} (6x)^3*7 = 4*1296x^3*7 = 36288x^3 \).
Step 6 ：The third term is \( \binom{4}{2} (6x)^2*7^2 = 6*1296x^2*49 = 380160x^2 \).
Step 7 ：The fourth term is \( \binom{4}{3} (6x)*7^3 = 4*6x*343 = 8232x \).
Step 8 ：The fifth term is \( 7^4 = 2401 \).
Step 9 ：Adding all these terms together, we get the expansion of \( (6x+7)^4 = 1296x^4 + 36288x^3 + 380160x^2 + 8232x + 2401 \).
Step 10 ：This is the simplest form of the expansion, so we have finished the problem.
Step 11 ：Finally, we check our result. The degree of the polynomial is 4, which is the same as the exponent in the original expression, so our result meets the requirements of the problem.
