Problem 18 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ \left(x^{2}+y\right)^{4} $$

Step-by-Step Solution

Answer

The simplified form of the binomial expression \( \left(x^{2}+y\right)^{4} \) is: \( x^{8} + 4x^{6}y + 6x^{4}y^{2} + 4x^{2}y^{3} + y^{4} \)

Step 1: Identification of Variables

In the expression \( \left(x^{2}+y\right)^{4} \), identify \( a = x^{2} \), \( b = y \), and \( n = 4 \)

Step 2: Application of the Binomial Theorem

The binomial theorem can be applied as follows: \( (x^{2}+y)^{4} = \sum_{k=0}^{4} {4 \choose k} (x^{2})^{4-k} * y^k \)

Step 3: Expansion of the binomial expression

Using binomial coefficient {(4 choose 0), (4 choose 1), (4 choose 2), (4 choose 3) and (4 choose 4)}, expand the expression like: \( \left(x^{2}+y\right)^{4} = 1x^{8} + 4x^{6}y + 6x^{4}y^{2} + 4x^{2}y^{3} + 1y^{4} \)