Problem 13 - Algebra for College Students

Question

Use the binomial theorem to expand each binomial. $$(a+2)^{3}$$

Step-by-Step Solution

Answer

(a+2)^3 = a^3 + 6a^2 + 12a + 8

Step 1: Identify the binomial

The binomial given is \((a+2)^{3} \). We will expand this using the binomial theorem.

Step 2: Understand the binomial theorem

The binomial theorem states that \((x + y)^n = \sum_{k=0}^{n}{n \choose k} x^{n-k} y^{k} \). Here, \(x = a \), \(y = 2 \), and \(n = 3 \).

Step 3: Write the general expanded form

Using the binomial theorem, \((a + 2)^3 \) expands to: \[ \sum_{k=0}^{3} \binom{3}{k} a^{3-k} \cdot 2^{k} \].

Step 4: Calculate the binomial coefficients

The binomial coefficients for \(n = 3 \) are \( \binom{3}{0} = 1 \), \( \binom{3}{1} = 3 \), \( \binom{3}{2} = 3 \), and \( \binom{3}{3} = 1 \).

Step 5: Apply the coefficients and simplify each term

For \(k=0 \): \[ \binom{3}{0} a^{3-0} 2^{0} = 1 \cdot a^{3} \cdot 1 = a^{3} \] For \(k=1 \): \[ \binom{3}{1} a^{3-1} 2^{1} = 3 \cdot a^{2} \cdot 2 = 6a^{2} \] For \(k=2 \): \[ \binom{3}{2} a^{3-2} 2^{2} = 3 \cdot a^{1} \cdot 4 = 12a \] For \(k=3 \): \[ \binom{3}{3} a^{3-3} 2^{3} = 1 \cdot a^{0} \cdot 8 = 8 \]

Step 6: Combine all terms

Adding all the terms from step 5 together: \[ (a+2)^{3} = a^{3} + 6a^{2} + 12a + 8 \]

Key Concepts

Binomial Expansion

Binomial expansion can seem complicated, but it helps in breaking down expressions like \( (a + 2)^3 \) into simpler terms.
The binomial theorem provides a structured way to expand expressions raised to a power, in this case, exponent 3.
The general form of the theorem is: \[ (x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k \].
Here, \( x = a, y = 2 \) and \( n = 3 \).
Using this theorem, we can systematically break down the expression into polynomial terms.
It makes handling larger exponents much easier.

Binomial Coefficients

Binomial coefficients are key to simplifying binomial expansions.
In our example \[ (a + 2)^3 \], each term in the expansion has a binomial coefficient, \(*\binom{n}{k}\)* based on the order of the term.
The binomial coefficient \(*\binom{n}{k}\)* is calculated as \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \].
For \( n = 3 \), the coefficients are: \[ \binom{3}{0} = 1, \binom{3}{1} = 3, \binom{3}{2} = 3, \binom{3}{3} = 1 \].
These coefficients help in predicting the multiplication factor for each term in the expanded form.

Polynomial Expansion

Polynomial expansion is the process of multiplying out expressions like \( (a + 2)^3 \) to express them as a polynomial.
The binomial theorem aids in this by organizing the steps to systematically expand the polynomial.
In our step-by-step solution, each term of the form \[ \binom{3}{k} a^{3-k} 2^k \] gives us a part of the overall polynomial.
The expanded form or polynomial for \( (a + 2)^3 \) ends up being \[ a^3 + 6a^2 + 12a + 8 \].
This is easier to work with compared to the binomial expression.

Algebraic Expressions

Understanding algebraic expressions is crucial when dealing with polynomials and the binomial theorem.
An algebraic expression is a combination of variables, coefficients, and constants.
For example, in the final term derived from \( (a + 2)^3 \), we have several algebraic expressions: \[ a^3 \], \[ 6a^2 \], \[ 12a \], and \[ 8 \].
Each of these has constants (numbers) and variables (letters like \( a \)).
Mastering algebraic expressions empowers you to tackle more complex problems efficiently.