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

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

Step 1: Recognize the given polynomial

The given polynomial is \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\). Observe the pattern of the coefficients \(1, 4, 6, 4, 1\), which resembles the binomial coefficients. This suggests the binomial expansion formula might be applicable.

Step 2: Identify as a binomial expansion

Recognize the polynomial as the expansion of a binomial expression. Specifically, these coefficients match the expansion of \((a+b)^4\), where the binomial theorem gives the expansion as:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] for \( n = 4 \) and \( inom{n}{k} = 1, 4, 6, 4, 1\) respectively.

Step 3: Set variables for the binomial expression

Determine the variables \(a\) and \(b\) in the expansion. Notice the powers of \(x\), which are decreasing, and the powers of \(y\), which are increasing. This suggests setting \(a = x^2\) and \(b = y\), giving the binomial expression \((x^2 + y)^4\).

Step 4: Confirm with binomial expansion

Verify that \((x^2 + y)^4\) expands to give the original polynomial. The expansion should yield:- \((x^2)^4 = x^8\)- \(4(x^2)^3y = 4x^6y\)- \(6(x^2)^2y^2 = 6x^4y^2\)- \(4(x^2)y^3 = 4x^2y^3\)- \(y^4\)All terms match exactly to the original polynomial.

Key Concepts

Binomial Coefficients

Binomial coefficients are essential in understanding how to expand powers of binomials. They are the numbers found in the expansion of a binomial raised to a power and are represented in Pascal's Triangle. For the binomial expansion \[(a + b)^n = inom{n}{0}a^n b^0 + inom{n}{1}a^{n-1} b^1 + inom{n}{2}a^{n-2} b^2 + ext{...} + inom{n}{n}a^0 b^n\]the coefficients \(inom{n}{k}\) are called binomial coefficients. They can be calculated using the formula: \[inom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(!\) denotes a factorial, the product of all positive integers up to that number.
Specifically, in our exercise, the polynomial \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\) shows coefficients \(1, 4, 6, 4, 1\), aligning with Pascal's Triangle for \(n = 4\). This suggests the polynomial is a binomial expansion of some expression, utilizing these coefficients.
Understanding binomial coefficients helps you recognize patterns within polynomials and predict how they expand when raised to a power.

Polynomial Factoring

Factoring polynomials simplifies the expressions by breaking them down into products of other simpler polynomials. In the original exercise, factoring involves recognizing the polynomial as a special form or pattern that can be rewritten more simply.
Notice that the given polynomial is structured with descending powers of \(x\) and ascending powers of \(y\), which indicates it might be a result of expanding a binomial expression.

• The polynomial \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\) is this special kind of expression, resulting from a binomial expression.
• To factor a polynomial this way, identify any common recognizable patterns that align with known binomial expansions.

In this instance, recognizing powers of \(x^2\) and \(y\) led to identifying the polynomial as the expansion of \((x^2 + y)^4\). This makes it significantly easier to work with and understand.

Binomial Expansion Formula

The Binomial Expansion Formula allows us to express powers of binomials as a sum of terms involving binomial coefficients. It is expressed as:\[(a + b)^n = \\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]where \(n\) is the exponent, and \(a\) and \(b\) are the terms being raised.
In practice, this means you multiply out an expression like \((x+y)^4\) using the coefficients from \(1, 4, 6, 4, 1\), derived from Pascal's Triangle, acting as weights for each term.

• These coefficients determine how each term in the expansion is constructed from the powers of \(a\) and \(b\).
• In our exercise, setting \(a = x^2\) and \(b = y\), allowed us to recreate the given polynomial by applying these coefficients to the appropriate powers of these terms.

By verifying the expansion step by step, you ensure the polynomial matches the original. This process demonstrates not only how binomials can factor polynomials but also how useful the binomial theorem is in solving algebraic problems.