Problem 8 - Beginning and Intermediate Algebra

Question

Use Pascal’s Triangle to expand each binomial. $$(k+2)^{5}$$

Step-by-Step Solution

Answer

Using Pascal's Triangle, we find the coefficients for the expansion of \((k+2)^5\). The expansion is: \((k+2)^5 = k^5 + 10k^4 + 40k^3 + 80k^2 + 80k + 32\)

Step 1: Use Pascal's Triangle to find the coefficients of the expansion

First, we need to find the coefficients for the expansion using Pascal's Triangle. Since we are considering a fifth-degree binomial, the row we need is the fifth row (counting from 0) of Pascal's Triangle. The coefficients in the fifth row are 1, 5, 10, 10, 5, and 1. These numbers will help us with the expansion.

Step 2: Use the coefficients to expand the binomial

Now that we have the coefficients from Pascal's Triangle, we will use them to expand the binomial \((k+2)^5\). Remember that the exponents on k will decrease from 5 to 0, while the exponents on 2 will increase from 0 to 5, and the coefficients from the triangle will be multiplied with each term. Our expansion will have the following terms: 1. \(\text{Coefficient: 1} \times k^5 \times 2^0 = k^5\) 2. \(\text{Coefficient: 5} \times k^4 \times 2^1 = 10k^4\) 3. \(\text{Coefficient: 10} \times k^3 \times 2^2 = 40k^3\) 4. \(\text{Coefficient: 10} \times k^2 \times 2^3 = 80k^2\) 5. \(\text{Coefficient: 5} \times k^1 \times 2^4 = 80k\) 6. \(\text{Coefficient: 1} \times k^0 \times 2^5 = 32\)

Step 3: Combine the results

Now, we will simply add all these terms together to form the expanded binomial: \((k+2)^5 = k^5 + 10k^4 + 40k^3 + 80k^2 + 80k + 32\) And that is the final expansion of the given binomial using Pascal's Triangle.

Key Concepts

Binomial Expansion

The binomial expansion is a method used to expand expressions that are raised to a power, in the form of \((a+b)^n\). This process allows us to express the power as a sum of terms involving coefficients, base variables, and their respective exponents. The binomial theorem provides a structured way to perform this expansion without actually multiplying the terms repeatedly. To facilitate the expansion,

• we find the coefficients using Pascal's Triangle,
• adjust the exponents of the terms accordingly,
• and form the expanded expression step by step.

For instance, the expression \((k+2)^5\) is expanded using binomials as follows:

• The exponent 5 indicates we need the coefficients from the fifth row of Pascal’s Triangle: 1, 5, 10, 10, 5, and 1.
• Next, apply these coefficients, decreasing the power of \(k\) from 5 to 0, and increasing the power of 2 from 0 to 5.

This approach prevents errors associated with manual multiplication and helps manage complex mathematical expressions effectively.

Polynomial Coefficients

Polynomial coefficients are numerical factors that multiply the variable terms in a polynomial expression. They determine the scale or weight of each term within an expression. These coefficients can be efficiently determined using Pascal’s Triangle, especially in the context of binomial expansion.

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two directly above it. In expanding the polynomial \((k+2)^5\), the coefficients are obtained from the entry row of Pascal's Triangle that corresponds to the power of the binomial. Here, we use the coefficients 1, 5, 10, 10, 5, and 1 from the fifth row. These numbers define:

• 1 as the coefficient of \(k^5\),
• 5 for \(k^4\),
• 10 for terms involving \(k^3\) and \(k^2\),
• another 5 for \(k\), and
• finally 1 for the constant term.

Using Pascal's Triangle ensures accuracy and makes the process of finding polynomial coefficients straightforward and reliable.

Algebraic Expressions

An algebraic expression is a mathematical statement comprising numbers, variables, and arithmetic operations. Understanding these expressions is crucial when dealing with polynomials and their expansions, as seen with binomials like \((k+2)^5\).

In any algebraic expression, variables such as \(k\) represent unknown quantities, while constants like 2 are fixed values. Within an expansion task like this:

• Each term is a product of variables and constants combined through multiplication,
• with coefficients multiplying these products to give distinct terms.

By acknowledging how elements of an algebraic expression interact, it becomes feasible to perform operations such as expansions and simplification. Ultimately, this understanding forms the groundwork for mastering more complex algebraic tasks.

Combining Like Terms

Combining like terms involves the process of simplifying algebraic expressions by merging terms that have identical variable parts. This helps to create a refined, simpler form of an expression after expanding it.

In the expansion of \((k+2)^5\), each term comprises a power of \(k\) and a constant. While all terms here are distinct due to different powers of \(k\):

• \(k^5\),
• \(10k^4\),
• \(40k^3\),
• \(80k^2\),
• \(80k\), and
• 32.

Simplifying by combining like terms primarily becomes relevant in more complex expressions where multiple terms share the same variable powers. However, even where no like terms exist to combine, knowing how to recognize and simplify when possible remains a vital algebraic skill.