Problem 54 - Beginning and Intermediate Algebra

Question

Find the indicated term of each binomial expansion. \((z+3)^{9} ;\) seventh term

Step-by-Step Solution

Answer

The seventh term of the binomial expansion \((z+3)^9\) is \(61236z^3\).

Step 1: Identify the Formula for the Binomial Expansion

We will use the binomial theorem formula to find the binomial expansion of \((z+3)^9\). The formula for the binomial expansion of \((a+b)^n\) is given as: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

Step 2: Plug in the Given Values into the Formula

We are looking for the seventh term in the expansion of \((z+3)^9\). Substitute a = z, b = 3, n = 9 and k = 6 in the above formula: Seventh term = \(\binom{9}{6} z^{(9-6)} 3^6\)

Step 3: Calculate the Combination Coefficient and Simplify the Result

We need to evaluate \(\binom{9}{6}\): \(\binom{9}{6} = \frac{9!}{6!(9-6)!} = \frac{9!}{6!3!}\) Evaluate the factorials and the combination coefficient: \(9! = 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 362,880\) \(6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720\) \(3! = 3 \cdot 2 \cdot 1 = 6\) \(\binom{9}{6} = \frac{362,880}{(720)(6)} = \frac{362,880}{4320} = 84\) Now, plug in the combination coefficient and simplify: Seventh term = \(84 z^3 3^6\) Seventh term = \(84z^3(729)\)

Step 4: Final Answer

Multiply 84 with 729: Seventh term = \(61236z^3\) So the seventh term of the binomial expansion, \((z+3)^9\), is \(61236z^3\).