Problem 10 - Precalculus

Question

In Exercises \(10-13,\) use Pascal's Triangle to expand the given binomial. $$ (x+2)^{5} $$

Step-by-Step Solution

Answer

\((x + 2)^5 = x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32\).

Step 1: Identify the Binomial Expansion Formula

The binomial theorem can be expressed as \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]In this exercise, \(a = x\), \(b = 2\), and \(n = 5\). The goal is to expand \((x + 2)^5\) using this formula.

Step 2: Find the Coefficients Using Pascal's Triangle

Generate the 5th row of Pascal's triangle: 1, 5, 10, 10, 5, 1. This row gives the coefficients for the expansion because the exponent \(n = 5\).

Step 3: Expand the Binomial

Using the coefficients from Pascal's triangle, write the expanded form:\((x + 2)^5 = 1x^5 + 5x^4(2) + 10x^3(2)^2 + 10x^2(2)^3 + 5x(2)^4 + 1(2)^5\).

Step 4: Simplify Each Term in the Expansion

Calculate each term individually:- \(1x^5 = x^5\)- \(5x^4(2) = 10x^4\)- \(10x^3(2)^2 = 40x^3\)- \(10x^2(2)^3 = 80x^2\)- \(5x(2)^4 = 80x\)- \(1(2)^5 = 32\).

Step 5: Write the Final Expanded Form

Combine all the simplified terms to provide the final expanded expression:\((x + 2)^5 = x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32\).

Key Concepts

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it in the previous row. Each row corresponds to the coefficients of the expansion of a binomial raised to a power. For instance, the 0th row is just 1, and the first few rows look like this:

• Row 0: 1
• Row 1: 1, 1
• Row 2: 1, 2, 1
• Row 3: 1, 3, 3, 1
• Row 4: 1, 4, 6, 4, 1
• Row 5: 1, 5, 10, 10, 5, 1

When dealing with polynomial expansions, Pascal's Triangle is handy for finding the coefficients of each term. Therefore, for the expansion of \((x + 2)^5\), we use the coefficients from the 5th row: 1, 5, 10, 10, 5, 1. These numbers determine how many of each term are combined in the expansion.

Binomial Theorem

The Binomial Theorem provides a method to expand expressions in the form of \((a + b)^n\), where \(n\) is a non-negative integer. This theorem states that \((a + b)^n\) can be expressed as the sum:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here, \(\binom{n}{k}\) is a binomial coefficient, representing the number of ways to choose \(k\) elements from \(n\) elements without regard to order. In our exercise, \(a = x\), \(b = 2\), and \(n = 5\). Therefore, the expansion of \((x + 2)^5\) is calculated using these binomial coefficients and applying them to each term. This allows us to systematically compute each term of the polynomial.

Polynomial Expansion

Polynomial Expansion is the process of expressing a polynomial that is in a compact form, such as a binomial raised to a power, as a sum of multiple terms. In essence, it involves expanding the power into a longer expression. Using the Binomial Theorem in our example, we expand \((x + 2)^5\) by calculating each term:

• Coefficient from Pascal: 1 × \(x^5\) for \(x^5\)
• Coefficient from Pascal: 5 × \(x^4\) × 2 for \(10x^4\)
• Coefficient from Pascal: 10 × \(x^3\) × \(2^2\) for \(40x^3\)
• Coefficient from Pascal: 10 × \(x^2\) × \(2^3\) for \(80x^2\)
• Coefficient from Pascal: 5 × \(x\) × \(2^4\) for \(80x\)
• Coefficient from Pascal: 1 × \(2^5\) for \(32\)

Adding these, we rewrite the expression as a sum of terms.

Exponents in Algebra

Exponents in Algebra are used to express repeated multiplication of a number or variable by itself. When working with binomial expansions, understanding how to handle exponents is crucial. In our exercise, we expanded \((x + 2)^5\). Here’s how it worked:

• \(x^5\) means \(x\) is multiplied by itself five times.
• In \(10x^4(2)^2\), \(x\) is raised to the power of 4, and \(2\) is squared.
• Every term contains \(x\) with exponents decreasing from 5 to 0 and \(2\) with exponents increasing from 0 to 5.

Manipulating exponents correctly ensures each term is calculated accurately, maintaining the integrity of the expression's expansion. Understanding the behavior of exponents as they distribute over a multiplication or addition in algebra is vital.