Problem 78 - Algebra 2

Question

Expand each binomial. $$ (2 a-b)^{7} $$

Step-by-Step Solution

Answer

\[(2a - b)^{7} = 128a^{7} - 448a^{6}b + 672a^{5}b^{2} - 560a^{4}b^{3} + 280a^{3}b^{4} - 84a^{2}b^{5} + 14ab^{6} - b^{7}\]

Step 1: Expand Binomial using Binomial Theorem

Let's expand the given binomial \( (2a - b)^{7} \) using the binomial expansion theorem. According to the theorem, \[(a - b)^{n} = \sum_{k=0}^{n} (-1)^{k} {n \choose k} a^{n-k} b^{k}\]Therefore, expanding \( (2a - b)^{7} \) will give:\[(2a - b)^{7} = {7 \choose 0} (2a)^{7}(-b)^{0} - {7 \choose 1}(2a)^{6}(-b)^{1} + {7 \choose 2}(2a)^{5}(-b)^{2} - {7 \choose 3}(2a)^{4}(-b)^{3} + {7 \choose 4}(2a)^{3}(-b)^{4} - {7 \choose 5}(2a)^{2}(-b)^{5} + {7 \choose 6}(2a)^{1}(-b)^{6} - {7 \choose 7}(2a)^{0}(-b)^{7}\]

Step 2: Simplify Terms

Next step is to simplify all the terms to get the final expanded form:\[128a^{7} - 448a^{6}b + 672a^{5}b^{2} - 560a^{4}b^{3} + 280a^{3}b^{4} - 84a^{2}b^{5} + 14ab^{6} - b^{7}\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a way to expand expressions that are raised to a power, like \((x + y)^n\). It uses coefficients known as binomial coefficients, which can be calculated using combinations: \({n \choose k}\). For example, in \((a - b)^{n}\), the theorem states that:
\[(a - b)^{n} = \sum_{k=0}^{n} (-1)^{k} {n \choose k} a^{n-k} b^{k}\]
Here, \((-1)^k\) accounts for the alternating signs as you expand a negative binomial, like \((2a-b)\) in this problem. This expansion allows you to write a polynomial that is equal to the binomial raised to a power by summing terms that have binomial coefficients, powers of the first term \((2a)\), and powers of the second term \((-b)\). It's a powerful tool for algebra and allows you to break down complex polynomial expressions.

Polynomial Expansion

Polynomial expansion refers to the process of expanding an algebraic expression that consists of two or more terms (polynomials) raised to a power. When you expand a polynomial like \((2a - b)^7\), you transform it into a single polynomial where each term represents a part of the expanded binomial.
In the expansion using the Binomial Theorem, each term in the polynomial will have contributions from both components of the original binomial. For instance, \(128a^7\) represents contributions purely from \(a\), whereas \(-448a^6b\) includes influences from both \(a\) and \(b\). As you can see, every expanded term reflects the combination of coefficients (calculated as binomial coefficients), base components, and their respective exponents.
When fully expanded, a polynomial obtained from a binomial raised to a power, in this case, consists of 8 terms, with coefficients derived from the binomial theorem's expansion method. This process of polynomial expansion is commonly needed in higher math for solving and simplifying expressions.

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers that represents the coefficients in a binomial expansion. It's a handy tool to quickly find the binomial coefficients \({n \choose k}\), which are crucial for expanding a binomial.
In the triangle, each number is the sum of the two directly above it. For instance, the third row \([1, 2, 1]\) corresponds to the expansion \((x + y)^2\). To use Pascal’s Triangle for \((2a - b)^7\), you’d look at the eighth row \([1, 7, 21, 35, 35, 21, 7, 1]\), as powers start counting from zero.
These numbers are the coefficients of the expanded form of the binomial, which is multiplied by the respective powers of the individual terms in the binomial. Understanding Pascal's Triangle not only simplifies determining coefficients but also connects to other mathematical patterns, such as combinations and probabilities.