Problem 39 - Precalculus: Mathematical for Calculus

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 as \((x+y)^{4}\).

Step 1: Recognize the Pattern

Observe that the given expression \(x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}\) appears to match the form of a binomial expansion. Specifically, it resembles \((x+y)^{4}\).

Step 2: Verify the Terms

We need to verify that the terms match the formula for the binomial expansion of \((x+y)^{4}\), which is \[\binom{4}{0} x^{4} y^{0} + \binom{4}{1} x^{3} y^{1} + \binom{4}{2} x^{2} y^{2} + \binom{4}{3} x^{1} y^{3} + \binom{4}{4} x^{0} y^{4}\]. Comparing each corresponding coefficient confirms the match.

Step 3: Use Binomial Theorem

Since the expression matches the form given in the Binomial Theorem, we can express the original polynomial as \((x+y)^{4}\) by observing the full sequence of coefficients \(1, 4, 6, 4, 1\).

Step 4: Conclude the Factorization

Having verified the arrangement of terms and coefficients, we conclude that the polynomial \(x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}\) is factored completely as \((x+y)^{4}\).

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of breaking down a complex polynomial into simpler components, or 'factors.' This helps to simplify expressions and solve polynomial equations more easily. When we factor a polynomial, we aim to express it as a product of its factors. The original polynomial can be thought of as a "product" that has been expanded, similar to turning ingredients into a finished meal.In the given exercise, the polynomial is factored using the Binomial Theorem, which reveals that the expression in question can be written as a power of a binomial, specifically \((x+y)^4\). This factored form is simpler and more compact. Factoring is especially helpful in solving equations where we're required to find roots or zeros of the polynomial. Once in factored form, understanding the individual factors opens the door to deeper insights into the polynomial's properties and potential solutions.

Binomial Expansion

Binomial expansion involves expanding expressions that are raised to a power, specifically polynomials with two terms. It is a specific case of polynomial expansion that uses a formula to expand expressions like \((x + y)^n\), where \(n\) is a non-negative integer.The Binomial Theorem provides the framework to expand any binomial. The formula is \[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k},\]where \(\binom{n}{k}\) represents binomial coefficients. These coefficients can be calculated using combinations, which tell us how many ways we can choose \(k\) elements from a set of \(n\) elements.By using binomial expansion, complex polynomials can be rewritten into more manageable forms. In our exercise, the transformation of the polynomial \(x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\) into \((x+y)^4\), illustrates the magic of using binomial expansion to simplify expressions.

Pascal's Triangle

Pascal's Triangle is a powerful tool that provides binomial coefficients in a triangular form, making calculations for binomial expansions quick and efficient. Each row in Pascal's Triangle corresponds to the coefficients of the expanded form of \((a + b)^n\). The triangle starts with a peak value of one at the top.To construct Pascal's Triangle:

• Begin with a "1" at the top.
• Each following row starts and ends with "1".
• The interior values are obtained by summing the two values directly above in the previous row.

Pascal's Triangle allows us to quickly find the binomial coefficients without computing combinations manually. For example, the fourth row, "1, 4, 6, 4, 1," corresponds to the coefficients needed for \((x+y)^4\), confirming the solution in our exercise. Understanding and using Pascal's Triangle can greatly simplify the process of binomial expansion and provide insights into polynomial patterns.