Problem 48 - College Algebra

Question

Find the term indicated in each expansion. \((x+2 y)^{10} ;\) the term containing \(y^{6}\)

Step-by-Step Solution

Answer

The term in the expansion containing \(y^{6}\) is \(13440x^4y^6\).

Step 1: Identify the Binomial Theorem

For every natural number \(n\), the binomial theorem states that \((x+y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k. \] The \({n \choose k}\) coefficient is the binomial coefficient which can be calculated using the formula \({n \choose k} = \frac{n!}{k!(n-k)!}\), where '!' denotes the factorial function.

Step 2: Identify the Term of Interest

Here, the term of interest is the one that contains \(y^{6}\). According to the binomial theorem, this term will be of the form \({10 \choose k} x^{10-k} (2y)^k\), where the power of \(y\) is 6. Therefore, \(k = 6\) in this case.

Step 3: Substitute the Values

Plug \(k = 6\) into the formula and also take into account that in this case \(y\) is actually \(2y\), the term becomes \({10 \choose 6} x^{10-6} (2y)^6\) = \(210x^4(64y^6)\ = 13440x^4y^6\).

Step 4: Final Answer

The term in the expansion of \((x+2 y)^{10}\) that contains \(y^{6}\) is \(13440x^4y^6\).