Problem 44 - Precalculus Mathematics for Calculus

Question

Factor using the Binomial Theorem. $$\begin{array}{l}(x-1)^{5}+5(x-1)^{4}+10(x-1)^{3} \\\\+10(x-1)^{2}+5(x-1)+1\end{array}$$

Step-by-Step Solution

Answer

The expression is factored as \(x^5\).

Step 1: Identify the form of the expression

First, notice that the expression is written in the expanded form of a binomial. It resembles the expansion of \((a+b)^n\). Here, we have \((x-1)\) in place of \(a\) and it forms coefficients akin to binomial coefficients.

Step 2: Recognize the binomial expansion pattern

The coefficients in the expression \(1, 5, 10, 10, 5, 1\) correspond to the binomial coefficients for \((a+b)^5\). The general term in the expansion of \((a+b)^n\) is \(\binom{n}{k}a^{n-k}b^k\), which provides these coefficients when \(n = 5\).

Step 3: Substitute back into the binomial theorem

Given the recognized pattern, we can substitute \((x-1)\) as \(a\) and constant \(1\) as \(b\) to reconstruct the binomial: \((a + 1)^5 = (x-1 + 1)^5 = x^5\).

Step 4: Verify the binomial construction

After determining that the binomial that expands to the given expression is \(x^5\), each term and coefficient matches the expanded expression when originally derived from \((x-1 + 1)^5 = x^5\). Thus the expression is completely factored as \(x^5\).

Key Concepts

Binomial Coefficients

In mathematics, binomial coefficients play an important role in various calculations involving binomials and combinations. These coefficients appear in the expanded form of binomial expressions like \((a + b)^n\). They are the numbers that make up Pascal’s Triangle, a simple arrangement where each entry is the sum of the two directly above it.For any given row in Pascal’s Triangle or a power \(n\), the binomial coefficient \(\binom{n}{k}\) refers to the coefficient of the term in the expansion of \((a + b)^n\). Here, it is calculated as follows:

• \(\binom{n}{0} = 1\)
• \(\binom{n}{1} = n\)
• \(\binom{n}{2} = \frac{n(n-1)}{2}\)
• Continuing this pattern, each coefficient is arrived at using previously counted coefficients.

The sequence \(1, 5, 10, 10, 5, 1\) perfectly illustrates binomial coefficients for the expansion of \((a+b)^5\), as shown in our expression.

Polynomial Expansion

The process of polynomial expansion involves transforming an expression into a form where it can be easily worked with or factored. Consider the expansion of a binomial, described by the Binomial Theorem: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)In simpler terms, each term in the expansion of \((a+b)^n\) is a combination of terms involving powers of \(a\) and \(b\), multiplied by their respective coefficients provided by the binomial coefficients.With regard to our given exercise, substituting \(a = x-1\) and \(b = 1\) for each term in the expansion gives us \( (x-1)^n\), yielding the original polynomial when expanded out this way. This pattern matches the original polynomial given.

Factoring Expressions

Factoring expressions is an essential skill in algebra that involves writing a polynomial or algebraic expression as the product of its factors.In our exercise, the expression is initially presented in expanded form but it has its roots in a factored binomial form as \((x-1+1)^5 = x^5\). Recognizing familiar patterns, such as binomial expansions, aids in quickly and accurately factoring complex expressions.Factoring allows for easier simplification and evaluation of algebraic expressions, as well as being crucial for solving equations and finding roots. In fact, factoring can bridge complex expressions down to simpler forms, making them more understandable and manageable.

Mathematical Proof

The ability to prove mathematical statements increases understanding and confidence in key principles. In the context of our problem, proving that the expanded polynomial expression matches the expected result involves clear steps and logical reasoning.The proof here relies on recognizing the coefficients and terms as originating from the binomial theorem:

• By identifying the sequence \(1, 5, 10, 10, 5, 1\), one concludes these are binomial coefficients for \((x)^5\).
• Substituting variables like \(x-1\) and checking each term ensures that the transformation holds.
• The verification step reinforces that the polynomial expansion matches \(x^5\) perfectly.

With the careful examination of each component, proof not only confirms correctness but deepens comprehension through logical analysis.

Algebraic Manipulation

Algebraic manipulation is the art of rewriting and transforming expressions without altering their values or relationships. It is the toolbox that allows mathematicians to solve equation puzzles, simplify expressions, and unravel mysteries hidden within.For our exercise, algebraic manipulation starts with the expression \((x-1)^5 + 5(x-1)^4 + 10(x-1)^3 + 10(x-1)^2 + 5(x-1) + 1\). Using algebraic manipulation techniques, we substitute and rearrange terms to find a simpler form.These techniques include:

• Substitution: Replacing expressions with equivalent terms to simplify working out solutions.
• Rearranging terms based on commonly identified patterns, like recognizing powers of binomials.
• Combining like terms to condense expressions.

Efficient algebraic manipulation aids in solving problems with speed and clarity, turning seemingly complicated expressions into easily digestible forms.