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

Step 1: Recognize the Pattern

Notice that the given polynomial is a special form which resembles the expansion of a binomial raised to the power of 4. Specifically, it looks similar to the expansion of \( (x+y)^4 \).This pattern suggests that we can apply the Binomial Theorem to factor it.

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \).For \( n=4 \), and by taking \( a=x \) and \( b=y \), the expansion becomes:\[ (x+y)^4 = \binom{4}{0}x^4y^0 + \binom{4}{1}x^3y^1 + \binom{4}{2}x^2y^2 + \binom{4}{3}x^1y^3 + \binom{4}{4}x^0y^4 \= x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4. \\]This matches the given polynomial.

Step 3: Confirm the Match

Verify that each term in the expanded form \( (x+y)^4 \) corresponds exactly to the terms in the original polynomial.Compare:1. \( x^4 \)2. \( 4x^3y \)3. \( 6x^2y^2 \)4. \( 4xy^3 \)5. \( y^4 \)These terms match perfectly, so the factorization is accurate.

Key Concepts

Polynomial Factorization

Polynomial factorization involves breaking down a complex polynomial into simpler polynomials that, when multiplied together, give back the original polynomial. This is useful for solving equations, simplifying expressions, and understanding the underlying structure of algebraic expressions. In many cases, recognizing patterns such as the complete square or difference of squares can simplify the factorization process.
For example, in polynomial expressions like \( x^4 + 4x^3 y + 6x^2 y^2 + 4xy^3 + y^4 \), it can be helpful to identify these patterns to factor efficiently. In this problem, the polynomial is a perfect square of a binomial raised to a power. Recognizing that it matches the expansion of \((x+y)^4\) allows us to directly factor the expression using the binomial theorem. This efficient shortcut highlights how understanding polynomial structures can tremendously simplify algebraic manipulation.

Binomial Expansion

The binomial expansion is a method for expanding expressions that are raised to a power. It involves using coefficients derived from Pascal's Triangle. The Binomial Theorem itself can be expressed as \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). This formula provides a systematic way to expand any binomial raised to any power, which is particularly useful when dealing with large exponents.
For instance, when expanding \((x + y)^4\), you strategically apply the formula to get each term:

• First term: \(\binom{4}{0}x^4y^0 = x^4\)
• Second term: \(\binom{4}{1}x^3y^1 = 4x^3y\)
• Third term: \(\binom{4}{2}x^2y^2 = 6x^2y^2\)
• Fourth term: \(\binom{4}{3}x^1y^3 = 4xy^3\)
• Fifth term: \(\binom{4}{4}x^0y^4 = y^4\)

This direct approach ensures that each term in the expansion corresponds perfectly to the terms in the original polynomial.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, and division) that represent a specific quantity or relationship. They form the foundation of algebra and allow us to model real-world situations with mathematical precision.
Expressions like \(x^4 + 4x^3 y + 6x^2 y^2 + 4xy^3 + y^4\) showcase how various terms with different powers and coefficients can be organized to represent complex quantities. These expressions are not static; they can be manipulated, factored, expanded, or simplified to reveal deeper insights or solve particular problems.
Understanding the behavior and properties of algebraic expressions enables us to effectively use mathematical tools like the binomial theorem or polynomial factorization, offering clear pathways to solutions.