Problem 35 - Algebra 2

Question

Simplify. $$ (3 b-c)^{3} $$

Step-by-Step Solution

Answer

\(27b^3 - 27b^2c + 9bc^2 - c^3\)

Step 1: Identify the Expression Type

The given expression is \[(3b-c)^3\]which is a binomial expression raised to the power of 3. We will expand it using the binomial theorem.

Step 2: Apply the Binomial Theorem

The binomial theorem states that \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]In our case, take \(a = 3b\), \(b = -c\), and \(n = 3\). Substitute these into the formula and expand.

Step 3: Calculating Each Term of the Expansion

Using the binomial theorem, calculate each term step by step:- First term: \(\binom{3}{0} (3b)^3 (-c)^0 = 27b^3\)- Second term: \(\binom{3}{1} (3b)^2 (-c)^1 = -27b^2c\)- Third term: \(\binom{3}{2} (3b)^1 (-c)^2 = 9bc^2\)- Fourth term: \(\binom{3}{3} (3b)^0 (-c)^3 = -c^3\).

Step 4: Combine All Terms

Write down the expression by combining all the calculated terms:\[27b^3 - 27b^2c + 9bc^2 - c^3\]This is the simplified form of the original expression.

Key Concepts

Algebraic expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Unlike equations, they do not contain an equals sign. They are a foundational element in algebra, allowing us to represent real-world situations in a mathematical form.
Algebraic expressions can be composed of a single term, such as \(3x\), or multiple terms combined through addition or subtraction, like \(3x + 2y - z\). Each term in an expression can have both coefficients (numerical part) and variables (the letter part).
The expression \((3b-c)^3\) is an example of a compound algebraic expression where two terms, \(3b\) and \(-c\), are combined using an exponentiation operation. Understanding how to manipulate such expressions is key in algebra, and involves techniques like expansion, simplification, and factoring.

Polynomial expansion

Polynomial expansion is the process of breaking down a power expression into a sum of simpler terms. It is often used in algebra to simplify expressions or solve equations.
The binomial theorem is a powerful tool for polynomial expansion. It gives a systematic method to expand expressions of the form \((a + b)^n\), where \(a\) and \(b\) are any terms, and \(n\) is a non-negative integer.
In polynomial expansion, each term of the expanded form can be calculated using the formula \(\binom{n}{k}a^{n-k}b^k\), where \(\binom{n}{k}\) is a binomial coefficient. By substituting the values into this formula, we can expand and simplify expressions like \((3b-c)^3\) into more manageable terms, each involving a product of powers of \(3b\) and \(-c\).

Binomial coefficients

Binomial coefficients are the numeric factors that multiply each term in the expansion of a binomial expression. They are denoted by \(\binom{n}{k}\), pronounced "n choose k," and are calculated using the formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Here, "!" represents the factorial operation, which means the product of all positive integers up to that number. For example, \(3! = 3 \times 2 \times 1 = 6\).
In the context of

• \((3b-c)^3\), the binomial coefficients are \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\). Each coefficient is multiplied by the corresponding powers of \(3b\) and \(-c\) to form the terms of the polynomial expansion.

These coefficients are crucial because they determine the multiplicative weight of each term in the expanded polynomial, directly affecting the final simplified expression.