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 factors to \((x^5)\).

Step 1: Recognize the Polynomial Form

Notice that the expression \[(x-1)^{5}+5(x-1)^{4}+10(x-1)^{3}+10(x-1)^{2}+5(x-1)+1\]is in the form of a binomial expansion. Each coefficient corresponds to the binomial coefficients, indicating that this is the expansion of \((x-1+1)^5\). The terms align with the binomial expansion of a power of a binomial form \((a + b)^n\).

Step 2: Identify the Binomial Theorem

According to the Binomial Theorem, the expansion of \((a+b)^n\) is given by: \[\sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k\],where \(\binom{n}{k}\) are the binomial coefficients.

Step 3: Apply the Binomial Theorem

Here, the expression is already expanded using the binomial theorem. Specifically, it comes from the expansion of \((x-1 + 1)^5 = x^5\).All combined binomial coefficients and terms match exactly with those in the given polynomial.

Step 4: Factor the Original Expression

Since the polynomial matches the expanded form from the Binomial Theorem, we can directly factor it into \((x^5 - 1)^1\),meaning the binomial can be written as \((x-1+1)^5 = x^5\).

Key Concepts

Binomial Expansion

The binomial expansion is a way to express a binomial expression that is raised to a power, in terms of simpler expressions. When we have a binomial like \[(a + b)^n,\] we can expand it into several terms rather than just leave it in its power form. This is especially useful when the exponent \(n\) is large. The expansion becomes more evident as a polynomial consisting of a sum of multiple terms. Each term in this expansion takes a specific form, deriving from the binomial theorem.Here's a simple breakdown of a binomial expansion:

• The binomial expression \((a + b)^n\) results in \(n + 1\) terms.
• Each term has a structure that includes the coefficients and the powers of \(a\) and \(b\).
• For example, \((x-1)^5\) is expanded as the series \[(x-1)^5 + 5(x-1)^4 + 10(x-1)^3 + 10(x-1)^2 + 5(x-1) + 1.\]

This expansion represents the full breakdown of a binomial raised by the power of 5, showcasing how different powers of the binomial's parts contribute to the resultant polynomial.

Binomial Coefficients

Binomial coefficients are the numerical factors that matter in the expansion of a binomial expression raised to any power. These coefficients are central to the binomial theorem, represented by the symbol \(\binom{n}{k}\), which is pronounced as "n choose k". This notation gives us a way to determine precisely how many terms we will see in each segment of the expanding polynomial.The calculation of binomial coefficients involves combinations, calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Where \(!\) denotes the factorial operation, meaning the product of all positive integers up to that number. Binomial coefficients are symmetrical;

• This means \(\binom{n}{k} = \binom{n}{n-k}\).
• They tell us how many ways we can arrange \(k\) objects out of \(n\), hence the association with choosing or combinations.

Within the binomial expansion of \[(x-1)^5,\] you can spot binomial coefficients as the numbers: 1, 5, 10, and so on, telling us the number of ways each term in the binomial is formed.

Polynomials

Polynomials are algebraic expressions that involve sums of power terms. Each of these terms can be comprised of constants, variables raised to a power, and coefficients. They are fundamental in expressing relationships within algebra.In the world of binomial theorems, any expanded binomial is ultimately a polynomial expressed with varying powers and terms.Polynomials have several key characteristics:

• They consist of terms where variables are raised to non-negative integers.
• The degree of a polynomial is the highest power of any single term's variable strength.
• Polynomials can be simple like \(x^2 + 3x + 2\), or complex like the expanded form of \((x-1)^5\).

The example given in \[(x-1)^5 + 5(x-1)^4 + 10(x-1)^3 + 10(x-1)^2 + 5(x-1) + 1\]is a polynomial that reflects the expanded version of the binomial, showing a beautiful harmony in the repeated structured pattern of algebra.