Problem 43 - Algebra and Trigonometry

Question

Factor using the Binomial Theorem. $$ x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+y^{4} $$

Step-by-Step Solution

Answer

The expression factors to \((x+y)^4\).

Step 1: Recognize the Expression Structure

First, look at the given expression \(x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+y^{4}\). Notice that it follows the pattern of a binomial expansion.

Step 2: Identify the Binomial Form

This expression resembles the expansion of \((x+y)^4\) using the Binomial Theorem, which states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Verify each term: \(\binom{4}{0}x^4y^0,\ \binom{4}{1}x^3y^1,\ \binom{4}{2}x^2y^2,\ \binom{4}{3}x^1y^3,\ \binom{4}{4}x^0y^4\).

Step 3: Verify the Binomial Coefficients

Assess the coefficients for each term: \(1, 4, 6, 4, 1\). These coefficients match the binomial coefficients of \((x+y)^4\) which are calculated as \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), \(\binom{4}{3} = 4\), \(\binom{4}{4} = 1\).

Step 4: Confirm and Write the Factorized Expression

By confirming that the structure and coefficients match, recognize that the expression \(x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+y^{4}\) is indeed the expansion of \((x+y)^4\). Thus, the factorized form is \((x+y)^4\).

Key Concepts

Factorization

Factorization is the mathematical process of breaking down an expression into simpler "factors" that, when multiplied together, give the original expression. This is like taking a complex puzzle and finding all the smaller pieces that make it up. In the exercise, the expression \(x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + y^{4}\) is factorized into \((x+y)^4\) using the concept of binomial expansion. To factorize, you need to observe the structure of the expression to find a common form. Once you recognize that the terms share similarities, such as the presence of powers of \(x\) and \(y\) in a symmetrical pattern, you can apply the Binomial Theorem to simplify the expression into its factorized form. This process allows you to understand the deep connection between the terms and their unified construction.

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power. Using the Binomial Theorem, expressions like \((a+b)^n\) can be expanded into a series of terms. Binomial expansion helps in finding out how each part of a binomial expression contributes to the whole, by expanding it into simpler addends. In the given problem, the original polynomial \(x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + y^{4}\) is an example of an expanded binomial of the form \((x+y)^4\). Each term follows the structure dictated by the theorem, which involves different combinations of powers of \(x\) and \(y\), multiplied by binomial coefficients. Understanding binomial expansion allows you to both expand and later, if required, simplify back into its compact binomial form.

Binomial Coefficients

Binomial coefficients are the numerical values that multiply the terms in a binomial expansion. They represent how many ways you can choose elements within a set, often denoted as \( \binom{n}{k} \), which is read as "n choose k". These coefficients play a vital role in determining the contribution of each term in an expanded binomial expression. For example, in the polynomial given in the exercise, the coefficients 1, 4, 6, 4, and 1 correspond to \( \binom{4}{0} \), \( \binom{4}{1} \), \( \binom{4}{2} \), \( \binom{4}{3} \), and \( \binom{4}{4} \) respectively. Each coefficient indicates the number of ways to combine \(x\) and \(y\) terms of certain degrees in the expansion. They can be quickly calculated using Pascal's triangle or the formula involving factorials: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). By understanding and using binomial coefficients, you can expertly navigate through binomial expansions and make sense of the expression's structure.