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Magazine Chapter 12: Problem 25
Expand and (where possible) simplify the expression. $$(a-b)^{5}$$
Short Answer
Expert verified
Question: Expand and simplify the expression \((a-b)^5\).
Answer: \(a^{5} - 5a^{4}b + 10a^{3}b^{2} - 10a^{2}b^{3} + 5ab^{4} - b^{5}\).
Step by step solution
01
Substitute values into the binomial theorem formula
We will substitute \(n=5\), \(a=a\), and \(b=-b\) into the binomial theorem formula:
$$(a-b)^{5} =\sum_{k=0}^{5} \binom{5}{k}a^{5-k}(-b)^{k}$$
02
Expand the expression
Now, we will expand the expression by evaluating each term in the sum:
\begin{align*}
(a-b)^5 &= \binom{5}{0}a^{5}(-b)^{0} + \binom{5}{1}a^{4}(-b)^{1} + \binom{5}{2}a^{3}(-b)^{2}\\
&\quad + \binom{5}{3}a^{2}(-b)^{3} +\binom{5}{4}a^{1}(-b)^{4} + \binom{5}{5}a^{0}(-b)^{5}\\
\end{align*}
03
Calculate binomial coefficients and simplify each term
We will now calculate the binomial coefficients (using the combination formula) and simplify each term:
\begin{align*}
(a-b)^5 &= \binom{5}{0}a^{5}(1) + \binom{5}{1}a^{4}(-b) + \binom{5}{2}a^{3}(b^{2}) \\
&\quad + \binom{5}{3}a^{2}(-b^{3}) +\binom{5}{4}a(-b)^{4} + \binom{5}{5}(-b)^{5}
\end{align*}
Calculating the binomial coefficients:
\begin{align*}
(a-b)^5 &= 1a^{5}(1) + 5a^{4}(-b) + 10a^{3}(b^{2}) \\
&\quad + 10a^{2}(-b^{3}) + 5a(-b)^{4} + 1(-b)^{5}
\end{align*}
04
Final simplified expression
Finally, we can write the expanded and simplified expression:
$$(a-b)^5 = a^{5} - 5a^{4}b + 10a^{3}b^{2} - 10a^{2}b^{3} + 5ab^{4} - b^{5}$$
So, the expanded and simplified expression of \((a-b)^{5}\) is \(a^{5} - 5a^{4}b + 10a^{3}b^{2} - 10a^{2}b^{3} + 5ab^{4} - b^{5}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial expansion
The binomial expansion is a fundamental concept in algebra that involves expressing a power of a binomial like \((a-b)^5\) as a sum of terms. Each term in this sum has a specific structure defined by the Binomial Theorem. Here is how it works:
- The binomial expression \((a-b)^{5}\) is expanded by breaking it into a series of simplified terms.
- The Binomial Theorem states that \((a+b)^n\) can be expanded as \(\sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^{k}\).
- For \((a-b)^5\), substitute \(-b\) for \(b\) to get the correct formula.
Combination formula
The combination formula, also known as the binomial coefficient, plays a critical role in calculating the coefficients in a binomial expansion. The formula is given by \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) stands for "n factorial," the product of all positive integers up to \(n\).
- In the formula, \(n\) represents the total number of items, while \(k\) represents the number of items to choose.
- Applying this to our exercise, the binomial coefficients are calculated for \((a-b)^5\).
- For example, \(\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10\).
Polynomial simplification
After expanding a binomial expression, the next step is polynomial simplification. This process involves reducing and combining like terms to achieve the simplest form of the expression.
- Each term of the binomial expansion considers the coefficients, resulting in terms like \(10a^3b^2\).
- During simplification, recognize and combine terms that have the same base and exponent values, though in this case, each term is unique in their power of \(a\) and \(b\).
- The goal is to clearly present the expanded polynomial in its simplest, most informative form.
Algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations that collectively express a mathematical rule or relationship. They are fundamental in representing relationships and in solving equations.
- In our exercise, \((a-b)^5\) is an algebraic expression representing a binomial raised to a power.
- To work with algebraic expressions efficiently, familiarity with operations such as expansion, factorization, and simplification is crucial.
- They encompass constants, variables, and terms. Operations can range from basic addition to complex factoring.
