Problem 29 - College Algebra

Question

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(2 a+b)^{6}$$

Step-by-Step Solution

Answer

The simplified form of \((2a+b)^6\) is \(64a^6 + 192a^5b + 240a^4b^2 + 160a^3b^3 + 60a^2b^4 + 12ab^5 + b^6\)

Step 1: Identify the components

In the equation \((2a+b)^6\), 'a' and 'b' are the components of the binomial and 'n' is the exponent, which is 6 in this case.

Step 2: Apply Binomial Theorem

Applying the Binomial Theorem, the equation \((2a+b)^6\), can be expanded as \(\sum_{k=0}^{6} {6 \choose k} (2a)^{6-k} b^{k}\). This gives a total of 7 terms (from k=0 to k=6).

Step 3: Calculate each term

Now, calculate each term individually:1. k=0: \( {6 \choose 0} (2a)^6 b^0 = 64a^6 \)2. k=1: \( {6 \choose 1} (2a)^5 b^1 = 192a^5*b \)3. k=2: \( {6 \choose 2} (2a)^4 b^2 = 240a^4*b^2 \)4. k=3: \( {6 \choose 3} (2a)^3 b^3 = 160a^3*b^3 \)5. k=4: \( {6 \choose 4} (2a)^2 b^4 = 60a^2*b^4 \)6. k=5: \( {6 \choose 5} (2a)^1 b^5 = 12a*b^5 \)7. k=6: \( {6 \choose 6} (2a)^0 b^6 = b^6 \)

Step 4: Write out the simplified binomial expansion

Combining all the calculated terms, the simplified form of the original binomial \((2a+b)^6\) is \(64a^6 + 192a^5b + 240a^4b^2 + 160a^3b^3 + 60a^2b^4 + 12ab^5 + b^6\)

Key Concepts

Binomial Expansion

The Binomial Theorem provides a quick way to expand or 'unpack' an algebraic expression raised to a power, such as ewlineewlineewlineewline (2a+b)^6. This theorem states that any binomial ewlineewlineewlineewline (a+b)^newlineewline can be expanded into the sum of terms involving 'a' and 'b' raised to various powers. The general form of the binomial expansion is given by ewlineewlineewlineewline ewlineewlineewlineewline ewlineewlineewlineewline ewlineewlineewlineewline ewlineewlineewlineewline ewlineewlineewlineewline ewlineewlineewlineewline ewlineewlineewlineewline ewlineewline. To use this expansion method, we identify 'a' and 'b' as the components of the binomial, and 'n' as its exponent.

We then create a series of terms starting from ewlineewlineewlineewline (a+b)^n = ewlineewlineewlineewline {6 ewlineewlineewlineewline choose ewlineewlineewlineewline k} a^{n-k} b^k, where 'k' ranges from 0 to n. Each term in this expansion can be calculated using combinations of 'n' choose 'k' to determine its coefficient.

Through this process, we transform a compact algebraic expression into a sum of simpler, individual terms, which can be easier to work with in algebraic manipulations.

Simplified Binomial Form

Once the binomial is expanded using the Binomial Theorem, we often need it in a simplified form that consolidates all the terms into a more digestible equation. Simplifying the binomial form involves calculating and simplifying each term of the expansion and then combining them. For each term, we apply the 'n choose k' formula for the coefficients and multiply by the appropriate powers of 'a' and 'b'.

In the case of ewlineewlineewlineewline (2a+b)^6, next would be the simplification of each term obtained from the theorem such as ewlineewlineewlineewline {6 ewlineewlineewlineewline choose ewlineewlineewlineewline 1} (2a)^5 b^1 or ewlineewlineewlineewline {6 ewlineewlineewlineewline choose ewlineewlineewlineewline 3} (2a)^3 b^3. To simplify, we calculate each binomial coefficient, perform the exponentiation, and multiply the results.

This gives us individual terms which are added to produce the final expanded and simplified binomial expression. Knowing how to obtain and simplify these terms is crucial to working effectively with polynomial expressions in algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like add, subtract, multiply, and divide) to represent quantities and relationships. The expression ewlineewlineewlineewline (2a+b)^6 is an example of a higher-degree polynomial, a type of algebraic expression with multiple terms.

Understanding the Binomial Theorem helps us to expand and simplify these polynomials, breaking them down into terms that are easier to work with. In algebra, much of the work involves manipulating these expressions to solve equations or to simplify complex formulae.

Polynomials are especially important because they appear frequently in different areas of mathematics and science, such as calculus, number theory, and physics. The skill of working with algebraic expressions, including expanding and simplifying them, is foundational for success in any advanced math course.