Problem 43 - Precalculus Mathematics 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 to \((x+y)^4\).

Step 1: Recognize the Pattern

The expression given is similar to the expansion of \((x + y)^4\). The Binomial Theorem tells us that \((x + y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k\). For \(n = 4\), the expansion is \((x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\).

Step 2: Verify Each Term

Match each term from the given expression with the corresponding term from \((x + y)^4\). The terms \(x^4, 4x^3y, 6x^2y^2, 4xy^3,\) and \(y^4\) all match exactly with the expansion of \((x + y)^4\).

Step 3: Confirm the Factorization

Since each term matches the binomial expansion, conclude that the given expression is indeed \((x + y)^4\). Write the factorized form as \((x+y)^4\).

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. This is similar to breaking down a number into its prime factors. By factorizing polynomials, we can simplify complex expressions, solve equations more easily, and understand the properties and roots of the polynomial.

Factors are simpler polynomials that, when multiplied together, reconstruct the original polynomial. For example, the polynomial \( x^2 - 5x + 6 \) can be factorized into \( (x - 2)(x - 3) \). This means that if we multiply \( (x - 2) \) and \( (x - 3) \), we get back \( x^2 - 5x + 6 \).

• Polynomial factorization helps in solving polynomial equations by setting each factor equal to zero and finding the roots.
• It is valuable in simplifying expressions, which can make computation or integration simpler in calculus.

Understanding polynomial factorization is key when working with any complex algebraic expressions, as it allows for breaking down into more manageable parts.

Binomial Expansion

The Binomial Expansion involves expanding expressions raised to a power, such as \((x + y)^n\). This results in a sum of terms of the form \( {n \choose k} x^{n-k} y^k \). The coefficients \( {n \choose k} \) are known as binomial coefficients and can be found in Pascal's Triangle.

For example, expanding \( (x + y)^4 \) gives us \( x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4 \). Each term corresponds to a particular combination of powers of \( x \) and \( y \), guided by the binomial theorem.

• Binomial expansion is useful in probability and statistics where binomial distributions are found.
• It's also pivotal in calculus, particularly in approximations and expansions.

The ability to rewrite expressions using binomial expansion simplifies many algebraic operations and reveals symmetries.

Power of a Binomial

The power of a binomial refers to expressing a binomial raised to an integer power, like \( (x + y)^n \). Using the Binomial Theorem, this can be expanded into a series of terms, each involving powers of \( x \) and \( y \), and the coefficients determined by combinations.

The power of a binomial reveals patterns and symmetries in polynomials that can help simplify calculations. The symmetrical property of the binomial coefficients—a key element in binomial expansion—ensures that the number of terms in the expansion follows a predictable structure.

• Understanding the power of a binomial aids in simplifying high polynomial powers, using their structure to find specific coefficients without full expansion.
• This concept is frequently used in combinatorics, calculus, and algebra.

Taking powers of binomials forms a backbone for many mathematical techniques, from solving equations to modeling phenomena in physics and statistics.