Problem 40 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

\(x^5\).

Step 1: Recognize the Binomial Theorem

The binomial theorem states that for any positive integer \(n\), \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). The given expression \((x-1)^5 + 5(x-1)^4 + 10(x-1)^3 + 10(x-1)^2 + 5(x-1) + 1\) resembles the expansion of \((a+b)^n\), where each term corresponds to the binomial coefficients for \(n = 5\).

Step 2: Identify Constants and Binomial Coefficients

Notice the coefficients 1, 5, 10, 10, 5, and 1 in the polynomial, which match the coefficients in the expansion of \((a+b)^5\). We can think of this polynomial as having been expanded from \((a+b)^5\) with \(a = x - 1\) and \(b = 1\).

Step 3: Observe Pattern for Complete Binomial Expansion

Given the coefficients and the powers of \((x-1)\), the complete polynomial can be rewritten directly from the binomial expansion as \([(x-1) + 1]^5\). This shows the entire expression fits the pattern \((a+b)^n\).

Step 4: Simplify the Binomial Expression

Substituting back into the binomial expansion, we simplify \([(x-1) + 1]^5\) to \(x^5\). This confirms the initial complex expression is a straightforward binomial.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that involves breaking down a complex polynomial into a product of simpler polynomials or factors. This process can help simplify expressions and solve equations more easily. In the context of the given exercise, factoring involves recognizing patterns that fit known expansions, such as those provided by the Binomial Theorem.
When you look at a polynomial like \\[ (x-1)^5 + 5(x-1)^4 + 10(x-1)^3 + 10(x-1)^2 + 5(x-1) + 1 \]
you can see it is structured in a specific way. It resembles a binomial expansion, meaning it was originally a simpler expression raised to a power. This understanding allows us to express it as a simpler factor, \[(x-1 + 1)^5 = x^5.\]
Recognizing this pattern is key to effective polynomial factoring.

Binomial Expansion

Binomial expansion refers to expanding a power of a binomial expression, such as \((a+b)^n\), into a sum of terms using binomial coefficients. The Binomial Theorem provides a formulaic approach to expanding these expressions.
For instance, if you have \((x-1)^5\), the expansion results in multiple terms with specific coefficients determined by the theorem. You observe:

• The polynomial \((x-1)^5\) expands to include terms with progressively decreasing powers of \((x-1)\) starting from 5 down to 0.
• Each of these terms involves coefficients which are calculated using factorial formulas, as outlined by the Binomial Theorem.

In our example, the expanded polynomial of \((x-1)^5\) is \((x-1)^5 + 5(x-1)^4 + 10(x-1)^3 + 10(x-1)^2 + 5(x-1) + 1\).
Identifying the expansion helps us combine the simplified form into a lone power expression \((a+b)^5\).

Binomial Coefficients

Binomial coefficients are integral to understanding binomial expansions. These coefficients are the numbers that multiply the terms in the binomial expansion, such as in \((a+b)^n.\) They are not arbitrary and follow a defined calculation.
The binomial coefficients are derived from the formula:\[binom{n}{k} = \frac{n!}{k!(n-k)!}.\]
This formula uses factorials to determine the specific coefficients for each term in the expansion. For example, for \((x-1)^5\), the coefficients are 1, 5, 10, 10, 5, and 1.

• The first term \((x-1)^5\) has a coefficient of 1.
• For the second term, as you decrease the power of \((x-1)\), you apply the coefficient 5, and so on for each subsequent term.

These coefficients ensure each term combines correctly to expand back into the original expression from its simpler form, making the computation of polynomial expressions straightforward once the pattern is recognized.