Problem 40 - Algebra and Trigonometry

Question

39 \(-42\) . 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 simplifies directly to \(x^5\).

Step 1: Recognize the Binomial Expansion

The expression given is a full binomial expansion. Specifically, it is written as \( (a+b)^n \) where \( a = x-1 \), \( b = 1 \), and \( n = 5 \).

Step 2: Compare with Binomial Theorem Formula

According to the binomial theorem, \( (x-1+1)^5 = (x)^5 = x^5 \) expands as \( \sum_{k=0}^{n} \binom{n}{k} (x-1)^{k} \cdot 1^{n-k} \) where \( n=5 \). The coefficients match perfectly with the terms given: \( 1, 5, 10, 10, 5, 1 \).

Step 3: Apply Identity

Since the given polynomial perfectly corresponds to the expansion of \( (x-1+1)^5 \), it simplifies back to \( x^5 \). Therefore, the step in finding a factor is complete as this reveals the identity of the expansion.

Step 4: Conclude the Factoring

The expression is already factored completely as \((x - 1 + 1)^5\) or simply \(x^5\). Given it matches the binomial expansion of \((x)^5\), no further factoring is necessary.

Key Concepts

Binomial Expansion

The Binomial Expansion is a fundamental concept that arises when you're working with expressions of the form \((a + b)^n\). The binomial theorem gives us the expanded form of this expression. In our original exercise, we see an expression that looks recognizable as one derived from the binomial theorem:

• It starts with \((x-1)^5\).
• The coefficients of the terms are \(1, 5, 10, 10, 5, 1\).

These coefficients might not initially seem significant, but they directly match the coefficients you would find when expanding \((a+b)^5\) — known famously as binomial coefficients. These coefficients can be found using Pascal's Triangle or the combination formula \(\binom{n}{k}\), where \(n\) is the power and \(k\) is the term's position in the sequence, starting from zero.
Applying this theorem simplifies working with higher powers because it provides a systematic way to handle what could otherwise be unwieldy expansions. If an expression seems difficult to factor at first glance but fits the pattern of a binomial expansion, you've struck gold! It simplifies the task incredibly by revealing the underlying identity or direct product the expression represents.

Polynomial Expansion

When dealing with Polynomial Expansion, as with binomial expansions, we focus on multiplying terms to get a specific form. In our solved problem, the expansion was initially written in its long form, showing all terms of \((x - 1)^5\) multiplied out.

• This included terms like \(10(x-1)^3\).
• Each was individually calculated using the given coefficients.

This representation is common in polynomial algebra, where each step of multiplication is separately outlined to ensure all parts are addressed. By recognizing the roles each term plays, it becomes easier to manage the entire expression.
For students, polynomial expansion can be conceptually quite challenging, especially for large powers. The binomial coefficients ease this by offering a predictable pattern, significantly reducing error. Once you've successfully expanded the terms using the coefficients, combining them to understand and identify patterns or simplifying can become more straightforward.

Factoring Polynomials

Factoring Polynomials involves breaking down complex expressions into simpler ones that are easier to handle. In our exercise, factoring began by recognizing the structure as a result from the binomial theorem. Sometimes, factoring is not about breaking expressions into smaller multiplicative factors, but identifying identities that hold true for the expression. In practical problems, when you have an expression expanded from something like \((x-1)^5\), you can quickly realize it's already in its simplest form by regrouping terms. Essentially, you're 'reverse-engineering' the expansion—seeing it from the perspective of what original binomial expression and power could have created it.

• The original expression was discovered to be already factored as \((x-1+1)^5\), which simplifies to \(x^5\).
• Recognizing when an equation is completely factored saves time.

Being able to fact-check expansions against factorial identities or special product identities can significantly expedite problem-solving in algebra. With frequent practice, identifying these traits and shortcutting laborious calculations becomes intuitive.