Problem 82 - Algebra and Trigonometry

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 \(10x^{6}y^{4}\).

Step 1: Recognize the binomial form

The given expression, \(\left(x^{2}+y^{2}\right)^{5}\), is a binomial raised to the power of 5. It can be expanded using the binomial theorem. We are tasked to find a term with \(x^{4}\) which will come from \((x^{2})^{2}\).

Step 2: Use the binomial theorem

The binomial theorem states that \([a + b]^{n} = ∑^{n}_{k=0} \binom{n}{k} a^{n-k} b^{k}\), where \(a=x^{2}\), \(b=y^{2}\), \(n=5\), and \(\binom{n}{k}\) is the binomial coefficient. The term containing \(x^{4}\) will be when \(k=2\) since \((x^{2})^{2} = x^{4}\).

Step 3: Calculate the binomial coefficient

The binomial coefficient \( \binom{n}{k} \) is calculated by \( \frac{n!}{k!(n-k)!} \) . For \(k=2\), the coefficient is \( \binom{5}{2} = \frac{5!}{2!(5-2)!} = 10 \).

Step 4: Substitute into the binomial theorem

Substitute these values into the formula to find the term with \(x^{4}\). This gives us the term as \(\binom{5}{2}(x^{2})^{5-2}(y^{2})^{2} = 10x^{6}y^{4}\).