Problem 42 - Precalculus: Mathematical for Calculus

Question

Factor using the Binomial Theorem. $$x^{8}+4 x^{6} y+6 x^{4} y^{2}+4 x^{2} y^{3}+y^{4}$$

Step-by-Step Solution

Answer

The polynomial factors as \((x^2 + y)^4\).

Step 1: Identify the Type of Polynomial

The given polynomial is \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\). Notice that the powers of \(x\) and \(y\) form a symmetrical pattern. It resembles the expansion of a binomial raised to a power.

Step 2: Recognize the Binomial Theorem Pattern

Recall the Binomial Theorem: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). The coefficients \(1, 4, 6, 4, 1\) match the Pascal's Triangle row for \(n=4\), i.e., \((a+b)^4\).

Step 3: Decide on the Variables

Identify each term in the polynomial as following the pattern of \((x^2 + y)\): \(x^8 = (x^2)^4\), \(4x^6y = 4(x^2)^3y\), \(6x^4y^2 = 6(x^2)^2y^2\), etc. This shows the terms are generated by \((x^2 + y)^4\).

Step 4: Reconstruct the Binomial Expression

Since the pattern confirms that the expression \((x^2 + y)^4\) generates the original polynomial, you can rewrite \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\) as \((x^2 + y)^4\).

Key Concepts

Polynomial Factorization

Polynomial factorization is like breaking down a complex cake recipe into smaller parts. Instead of dealing with the entire cake at once, you look for smaller, simpler recipes that combine to create the original cake. The goal is to express a polynomial as a product of simpler polynomials. In our exercise, we're asked to factor the polynomial \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\). By examining its structure, we identified it as being the fourth power of the binomial \((x^2 + y)\). This is a more compact and simpler form that highlights the repeated pattern seen in the polynomial.

• Look for patterns or symmetry in the polynomial.
• Check if it matches any known expansions, like those from the Binomial Theorem.
• Rewrite the polynomial expression in terms of its factors.

Understanding polynomial factorization helps in solving equations, simplifying problems, and revealing underlying mathematical structures.

Pascal's Triangle

Pascal's Triangle is a useful tool when working with binomial expansions. It's a triangular array of numbers, where each number is the sum of the two numbers directly above it. This simple structure encodes the coefficients used in the Binomial Theorem. For example, the row corresponding to the exponent 4 is \(1, 4, 6, 4, 1\). In our given polynomial \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\), we see these exact coefficients, signaling its relation to the fourth power of a binomial.

• Each row in Pascal's Triangle corresponds to a power of a binomial.
• The coefficients tell us how the terms in the binomial expansion combine.
• Recognizing these coefficients allows us to quickly identify the binomial decomposition.

So, when you see coefficients like these in a polynomial, you can explore the corresponding binomial to simplify or solve your exercise.

Polynomial Symmetry

Polynomial symmetry can be a hint to factorizing or simplifying a polynomial. In this context, symmetry refers to a repeated pattern within the powers of the terms used in a polynomial. Take a look at our exercise: \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\). Notice how the powers of \(x\) decrease while the powers of \(y\) increase symmetrically. This pattern is a telltale sign of a binomial expansion.

• Symmetry can indicate some underlying binomial structure.
• It helps in recognizing familiar patterns that simplify the polynomial.
• Finding symmetry is essential for quick factorizations and understanding polynomial characteristics.

Embracing the concept of symmetry can make analyzing and working with polynomials much easier, especially when combined with tools like Pascal’s Triangle.