Problem 25 - Contemporary Precalculus

Question

Expand and (where possible) simplify the expression. $$(a-b)^{5}$$

Step-by-Step Solution

Answer

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 1: Substitute values into the binomial theorem formula

We will substitute \(n=5\), \(a=a\), and \(b=-b\) into the binomial theorem formula: $$(a-b)^{5} =\sum_{k=0}^{5} \binom{5}{k}a^{5-k}(-b)^{k}$$

Step 2: 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*}

Step 3: 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*}

Step 4: 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 \(a^{5} - 5a^{4}b + 10a^{3}b^{2} - 10a^{2}b^{3} + 5ab^{4} - b^{5}\).

Key Concepts

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.

Each term in the binomial expansion corresponds to a specific value of \(k\) in the general formula.This systematic approach allows us to expand complex binomials into more manageable polynomial expressions.

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, \(\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10\).

These coefficients determine the magnitude of each term in the binomial expansion, allowing us to construct each part of the expanded expression.

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.

Simplifying polynomials is essential in algebra as it turns complex expressions into actionable data, making further calculations or evaluations straightforward and free of clutter.

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.

Understanding and manipulating algebraic expressions is a core algebra skill, essential for tackling a broad range of mathematical problems in academia and beyond.