Problem 14 - Algebra 2

Question

Expand each power. $$ (m-a)^{5} $$

Step-by-Step Solution

Answer

The expanded form of \((m-a)^5\) is \(m^5 - 5m^4a + 10m^3a^2 - 10m^2a^3 + 5ma^4 - a^5\).

Step 1: Identify the binomial

The given problem is to expand the expression \((m-a)^5\). This is a binomial expression because it consists of two terms: \(m\) and \(a\).

Step 2: Use the Binomial Theorem

The binomial theorem states that \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\). In this case, set \(x = m\), \(y = -a\), and \(n = 5\).

Step 3: Calculate each term using binomial coefficients

For each term in the expansion \((m-a)^5\), calculate using \(\binom{5}{k} m^{5-k} (-a)^k\) for \(k = 0\) to \(5\): 1. \(k=0\): \(\binom{5}{0} m^5 (1) = m^5\)2. \(k=1\): \(\binom{5}{1} m^4 (-a) = -5m^4a\)3. \(k=2\): \(\binom{5}{2} m^3 (-a)^2 = 10m^3a^2\)4. \(k=3\): \(\binom{5}{3} m^2 (-a)^3 = -10m^2a^3\)5. \(k=4\): \(\binom{5}{4} m (-a)^4 = 5ma^4\)6. \(k=5\): \(\binom{5}{5} (-a)^5 = -a^5\)

Step 4: Summarize the expanded form

Combine all the terms from Step 3 to get the expanded expression: \( m^5 - 5m^4a + 10m^3a^2 - 10m^2a^3 + 5ma^4 - a^5 \)

Key Concepts

Binomial expansion

The binomial expansion is a process of expanding expressions that are raised to a power, specifically those comprising two terms. In our case, we have

• a binomial
  • $$ \textrm{(m-a)^5} $$

This means the expression consists of two terms, "m" and "-a". When we talk about expanding, we mean breaking down the expression into a sum of terms involving powers of the two components. Binomial expansion uses coefficients and involves the calculation of the power of each term separately.
Understanding the binomial expansion can be really useful in areas such as algebra, calculus, and many applied sciences because it provides a simplified form of expressions and makes further calculations easier.

Binomial coefficients

Binomial coefficients play a vital role in the binomial expansion. They are the numbers that determine the weight each term has in the expansion. In the expansion formula

• $$ \binom{n}{k} $$

These numbers can also be found in Pascal's Triangle, which is a handy tool for quickly finding binomial coefficients.
For instance, expanding

• $$ (m-a)^5 $$

requires using the binomial coefficients for n = 5, derived from Pascal’s Triangle or calculated as

• $$ \frac{n!}{k!(n-k)!} $$

The coefficients

• 1, 5, 10, 10, 5, and 1 correspond to each term.

These numbers help determine how many times each term in the binomial appears in the expanded form.

Polynomial expansion

When expressing a binomial raised to a power in expanded form, it becomes a polynomial. This is called polynomial expansion. Each term in this polynomial has coefficients, the variable part, and the exponents of these variables.
Let's consider the polynomial expansion of

• $$ (m-a)^5 $$

Each term in the expansion will involve different power combinations of "m" and "a". Therefore, the expanded form delivers several terms summed together, showing the detailed breakdown of the initial expression.
In the expansion of

• $$ (m-a)^5 = m^5 - 5m^4a + 10m^3a^2 - 10m^2a^3 + 5ma^4 - a^5 $$

each term represents different combinations of the powers of "m" and "a". Polynomial expansion is an excellent way of writing complex expressions in a simpler form that is easier to work with in computations.

Exponentiation

Exponentiation is the mathematical operation of raising one number, the base, to the power of another number, the exponent. In expressions like

• $$ (m-a)^5 $$

the exponent "5" indicates how many times the base

• (m-a)

is multiplied by itself. This operation is a core component of the expansion process.
Using exponentiation, we transform our expression into a sum of terms by distributing the power across the terms of the binomial. Each exponentiation step reduces the complexity and helps break down the expression into simpler forms, making it easier to compute or further manipulate in algebraic procedures.