Question

consider the binomial expression ((x + y)^{10}). use the binomial theorem to complete each statement. the (square) term of the expanded form of the given binomial expression is (45x^{8}y^{2}). the (square) term of the expanded form of the given binomial expression is (252x^{5}y^{5}). the (square) term of the expanded form of the given binomial expression is (10xy^{9}).

Explanation:

Step1: Recall binomial theorem

The binomial theorem states:
$$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^{k}$$
where the $(k+1)$-th term is $\binom{n}{k}x^{n-k}y^{k}$.

Step2: Solve for first term

For $45x^8y^2$, $n=10$, $k=2$.
Term number: $k+1=2+1=3$
Check coefficient: $\binom{10}{2}=\frac{10!}{2!(10-2)!}=\frac{10\times9}{2\times1}=45$

Step3: Solve for second term

For $252x^5y^5$, $n=10$, $k=5$.
Term number: $k+1=5+1=6$
Check coefficient: $\binom{10}{5}=\frac{10!}{5!(10-5)!}=252$

Step4: Solve for third term

For $10xy^9$, $n=10$, $k=9$.
Term number: $k+1=9+1=10$
Check coefficient: $\binom{10}{9}=\frac{10!}{9!(10-9)!}=10$

Answer:

The 3rd term of the expanded form of the given binomial expression is $45x^8y^2$.
The 6th term of the expanded form of the given binomial expression is $252x^5y^5$.
The 10th term of the expanded form of the given binomial expression is $10xy^9$.