Problem 54 - Contemporary Precalculus

Question

Use the Binomial Theorem to factor the expression. $$x^{4}-4 x^{3}+6 x^{2}-4 x+1$$

Step-by-Step Solution

Answer

Question: Using the Binomial theorem, factor the expression $$x^{4}-4 x^{3}+6 x^{2}-4 x+1$$. Answer: The factored expression is $$(x-1)^4$$.

Step 1: Identify the Binomial theorem pattern

In the expression $$x^{4}-4 x^{3}+6 x^{2}-4 x+1$$, notice a pattern in the coefficients: 1, 4, 6, 4, 1. These are the numbers from the binomial expansion of \((a+b)^4\). Using the Binomial theorem, we get: $$(a+b)^4 = \binom{4}{0}a^4 b^0 + \binom{4}{1}a^3 b^1 + \binom{4}{2}a^2 b^2 + \binom{4}{3}a^1 b^3 + \binom{4}{4}a^0 b^4$$ Generates the coefficients 1, 4, 6, 4, 1 as desired.

Step 2: Rewrite the expression

We want to rewrite the expression $$x^{4}-4 x^{3}+6 x^{2}-4 x+1$$ to match the Binomial theorem expansion format \((a+b)^4\). Let's try substituting \(a=x\) and \(b=-1\) and see if this works: $$(x-1)^4 = \binom{4}{0}x^4(-1)^0 + \binom{4}{1}x^3(-1)^1 + \binom{4}{2}x^2(-1)^2 + \binom{4}{3}x^1(-1)^3 + \binom{4}{4}x^0(-1)^4$$ Simplifying, we get $$(x-1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1$$

Step 3: Write the factored expression

Now, we found that the given expression $$x^{4}-4 x^{3}+6 x^{2}-4 x+1$$ can be factored as: $$(x-1)^4$$

Key Concepts

Polynomial Factoring

Polynomial factoring is an essential skill in algebra that involves rewriting an expression as a product of smaller expressions, called factors. It simplifies complex polynomials and reveals their roots or solutions. When dealing with polynomial equations, identifying factors allows you to solve them more easily.

Factors of a polynomial are the expressions that multiply together to form the original polynomial. In the case of the expression \(x^4 - 4x^3 + 6x^2 - 4x + 1\), the aim is to express it as a product using recognizable patterns like the Binomial Theorem.

Key methods:

• Recognize patterns and use identities like the difference of squares or sum of cubes.
• Apply synthetic division and long division to divide polynomials.
• Identify common factors and factor them out from each term.

This exercise involves recognizing the binomial pattern and rewriting the polynomial as \((x-1)^4\) which is a succinct expression of the original polynomial.

Binomial Expansion

The Binomial Theorem is a powerful tool used to expand expressions that are raised to a power. It's especially useful for recognizing and simplifying complex polynomial expressions using a predictive pattern based on coefficients.

The general form of the binomial expansion is:
\[ (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \]
In this form, \(\binom{n}{k}\) represents a binomial coefficient, which can be calculated using factorials. The formula provides a systematic way to expand expressions like \((x+y)^n\).

In the given problem, the expression \(x^4 - 4x^3 + 6x^2 - 4x + 1\) follows the binomial expansion pattern of \((x+y)^n\). By identifying \(a = x\) and \(b = -1\) in \((x-1)^4\), we can confirm the coefficients match: 1, 4, 6, 4, 1. This indicates a successful application of the Binomial Theorem to factor the expression.

Mathematical Patterns

Recognizing mathematical patterns plays a crucial role in solving algebraic problems efficiently. Patterns help us predict outcomes and streamline solutions by leveraging known identities and properties.

Key patterns you may encounter in algebra include:

• Arithmetic Patterns: Sequences where difference between terms is constant.
• Geometric Patterns: Sequences where ratio between terms is constant.
• Polynomial Patterns: Like Pascal's Triangle, often associated with binomial coefficients.

By observing the coefficients 1, 4, 6, 4, 1 in the expression \(x^4 - 4x^3 + 6x^2 - 4x + 1\), we immediately recognize the binomial pattern\((a+b)^4\).

This provides insight that the expression is the expansion of \((x-1)^4\). Patterns not only simplify the factoring process but also enhance understanding of algebraic structures, helping in identifying shortcuts and streamlined pathways to solve equations effectively.