Problem 35 - College Algebra with Modeling and Visualization

Question

Use Pascal's triangle to help expand the expression. $$ (2-x)^{5} $$

Step-by-Step Solution

Answer

The expanded expression is \(32 - 80x + 80x^2 - 40x^3 + 10x^4 - x^5\).

Step 1: Identify the Binomial Theorem

The binomial theorem tells us that \((a+b)^n\) can be expanded as \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, we have the binomial \((2-x)^5\). Thus, \(a=2\), \(b=-x\), and \(n=5\).

Step 2: Determine Pascal's Triangle Row

Pascal's triangle provides the binomial coefficients. For \(n = 5\), the row in Pascal's triangle corresponding to these coefficients is \([1, 5, 10, 10, 5, 1]\).

Step 3: Apply Pascal's Triangle to the Expansion

Using the coefficients \([1, 5, 10, 10, 5, 1]\), expand the expression:\[(2-x)^5 = \sum_{k=0}^{5} \binom{5}{k} (2)^{5-k} (-x)^k\].

Step 4: Expand Each Term of the Binomial

Compute each term separately:- For \(k=0\): \(1 \times 2^5 \times (-x)^0 = 32\).- For \(k=1\): \(5 \times 2^4 \times (-x) = -80x\).- For \(k=2\): \(10 \times 2^3 \times x^2 = 80x^2\).- For \(k=3\): \(10 \times 2^2 \times (-x)^3 = -40x^3\).- For \(k=4\): \(5 \times 2^1 \times x^4 = 10x^4\).- For \(k=5\): \(1 \times 2^0 \times (-x)^5 = -x^5\).

Step 5: Write the Final Expanded Expression

Combine all terms together:\((2-x)^5 = 32 - 80x + 80x^2 - 40x^3 + 10x^4 - x^5\).

Key Concepts

Pascal's Triangle

Pascal's Triangle is a simple yet powerful tool in mathematics. It is a triangular array of numbers that shows the coefficients in the binomial expansion of any expression in the form \((a+b)^n\). Each row in the triangle corresponds to the terms in the expansion. For example, the first row is simply \([1]\), representing \((a+b)^0\). The second row is \([1, 1]\), representing \((a+b)^1\).

• Every row starts and ends with 1.
• Each interior number is the sum of the two numbers directly above it from the previous row.

For expanding the binomial \((2-x)^5\), we refer to the sixth row (since the first row is considered \(n=0\)), which is \([1, 5, 10, 10, 5, 1]\). These numbers serve as the coefficients in our polynomial expansion.

Binomial Theorem

The Binomial Theorem is a key principle that allows us to expand expressions that are raised to a power, like \((a+b)^n\). According to this theorem, the expression can be expanded into a sum involving terms of the form \( \binom{n}{k} a^{n-k} b^k \).
This allows us to express the power \((a+b)^n\) as a series, simplifying the calculation of high powers by breaking it into manageable terms.

• \(a\) and \(b\) are any two terms in the binomial.
• \(n\) is the power to which the binomial is raised.
• \(k\) is an index that ranges from 0 to \(n\).

In our exercise, \((2-x)^5\), \(a\) is 2, \(b\) is \(-x\), and \(n\) is 5. This gives us a structured way to compute each term using the specified coefficients from Pascal's Triangle.

Polynomial Expansion

Polynomial Expansion is the process of expressing a binomial expression raised to a power as a polynomial. This means creating a string of single terms, each with coefficients, and no further binomial is inside the polynomial.
Using the coefficients determined by Pascal's Triangle and the Binomial Theorem, you multiply each term by appropriate powers of the constituent parts of the binomial.
For \((2-x)^5\), the result becomes a polynomial:

• 32 (the constant term)
• -80x (first degree term)
• 80x^2 (second degree term)
• -40x^3
• 10x^4
• -x^5

Each term is computed by applying the Binomial Theorem, involving sequential calculations that are simple to execute with the guidance of the coefficients.

Algebra

Algebra is the branch of mathematics dealing with variables and the rules for manipulating these variables in expressions. It provides the foundation for understanding and solving equations, including those that arise in polynomial expansions.
In algebra, the distribution of terms, application, and understanding of functions like those derived from the Binomial Theorem is crucial. Such expansions demonstrate algebraic structure and rules.

• Understanding order of operations is vital in navigating these expansions.
• Concepts like variables, coefficients, and powers come into play.

In the case of our binomial \((2-x)^5\), understanding how to expand it using algebraic rules ensures you combine terms correctly, maintaining their sign and power order through careful calculation.