Problem 72 - College Algebra

Question

Find the term in the expansion of \(\left(x^{2}+y^{2}\right)^{5}\) containing \(x^{4}\) as a factor.

Step-by-Step Solution

Answer

The term in the expansion of \(\left(x^{2}+y^{2}\right)^{5}\) that contains \(x^{4}\) as a factor is \(10x^{4}y^{2}\).

Step 1: Use of Binomial Expansion

Expand \((x^{2}+y^{2})^{5}\) using binomial theorem. This theorem dictates that \((a + b)^{n}=\sum _{k=0}^{n} \binom{n}{k} a^{n-k}b^{k}\). Here, \(a\) and \(b\) are \(x^{2}\) and \(y^{2}\) respectively and \(n=5\). With this theorem, all terms in the expansion can be constructed.

Step 2: Identify Needed Term

Since we need a term with \(x^{4}\) as a factor, we need \(x^{2}\) to be squared, implying \(a^{n-k} = (x^{2})^{2}\), hence \(n-k = 2\). This provides one value of \(k = n - 2 = 5 - 2 = 3\). Substituting this into the binomial expansion yields the required term, \(\binom{5}{3}x^{4}y^{2}\). By definition, \(\binom{5}{3}\) is equal to \(\frac{5!}{3!(5-3)!}\), which simplifies to 10.

Step 3: Multiply Result to Simplify

Multiply 10 by \(x^{4}\) and \(y^{2}\) to get the term needed, \(10x^{4}y^{2}\).