Problem 46 - Precalculus: Mathematics 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

\((x^2 + y)^4\)

Step 1: Identify the Pattern

The expression \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\) resembles the binomial expansion of \((x+y)^n\). Notice the powers of \(x\) decreasing and \(y\) increasing systematically.

Step 2: Recognize Expansion Coefficients

Compare the coefficients in the expression, \(1, 4, 6, 4, 1\), with the coefficients in the binomial expansion \((x+y)^4\) which are the binomial coefficients \(\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3}, \binom{4}{4}\). This indicates that the expression might be the expansion of a higher power of a binomial.

Step 3: Check Binomial Theorem Compatibility

According to the binomial theorem, \((a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\). Rewrite the expression as \((x^2+y)^4\) and verify if it aligns perfectly with the structure and coefficients.

Step 4: Verify the Factorization

Confirm by expanding \((x^2 + y)^4\): 1. \((x^2)^4 = x^8\)2. \(4(x^2)^3y = 4x^6y\)3. \(6(x^2)^2y^2 = 6x^4y^2\)4. \(4x^2y^3 = 4x^2y^3\)5. \(y^4 = y^4\)Each term corresponds to a term in the original expression, confirming that the factorization is correct.

Key Concepts

Binomial Expansion

Binomial Expansion is a powerful tool in algebra that allows us to express polynomials in expanded form. Specifically, when a binomial expression like \((a+b)^n\) is expanded, it turns into a sum of terms involving powers of \(a\) and \(b\). Each term in the expansion of a binomial has a specific structure and consists of:

• A coefficient, which is a binomial coefficient.
• Powers of \(a\) and \(b\) that add up to \(n\).

To better understand this concept, let's consider the example given: \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\). This polynomial can be expressed as \((x^2 + y)^4\).
Notice how the powers of \(x^2\) and \(y\) are changing. They follow a specific sequence as per the binomial theorem, where \(x^2\) decreases from \(8\) to \(0\) (as \((x^2)^n\)), while \(y\) increases from \(0\) to \(4\). In every term, the sum of the exponents for \(x^2\) and \(y\) always equals \(4\), maintaining the balance of the binomial expansion.

Binomial Coefficients

Binomial Coefficients play a vital role in binomial expansions. These coefficients determine how terms are distributed and are derived from the Pascal's Triangle or calculated using combinations (binomial coefficients formula).
The formula for binomial coefficients is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(!\) denotes the factorial function.
In the polynomial \((x^2 + y)^4\), the coefficients \(1, 4, 6, 4, 1\) are obtained from \(\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3}, \binom{4}{4}\) respectively.

• \(\binom{4}{0} = 1\)
• \(\binom{4}{1} = 4\)
• \(\binom{4}{2} = 6\)
• \(\binom{4}{3} = 4\)
• \(\binom{4}{4} = 1\)

These coefficients are used to multiply each term in the expansion, ensuring that the distribution of terms is accurate and the expanded form matches what the Binomial Theorem predicts.

Polynomial Factorization

Polynomial Factorization involves breaking down a polynomial into simpler "factors" that, when multiplied together, will give the original polynomial.
In this exercise, we employed the Binomial Theorem's structure to identify and factor \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\) as \((x^2 + y)^4\).
Factoring polynomials requires recognizing patterns. Here, the pattern resembles the expansion of a binomial power due to:

• The systematic changes in powers of terms.
• The recognizable coefficients of a binomial expansion.

To confirm successful factorization, we re-expand the factor \((x^2 + y)^4\) and ensure its terms match the original polynomial, a crucial step in verifying the factorization's accuracy. Thus, polynomial factorization helps simplify complex expressions into basic, more manageable pieces and often reveals insightful mathematical properties.