Problem 72 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(3 x-2)^{3}$$

Step-by-Step Solution

Answer

The expanded form of \((3x - 2)^3\) is \(27x^3 - 54x^2 + 36x - 8\).

Step 1: Identify the pattern or formula

Recognize that we need to expand the expression \((3x - 2)^3\). In this case, we can use the formula for the cube of a binomial: \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\). Here, \(a = 3x\) and \(b = 2\).

Step 2: Apply the formula for a cube of a binomial

Substitute \(a = 3x\) and \(b = 2\) into the formula \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\).Let's calculate each term one by one:1. \(a^3 = (3x)^3 = 27x^3\)2. \(3a^2b = 3(3x)^2 \cdot 2 = 3 \cdot 9x^2 \cdot 2 = 54x^2\)3. \(3ab^2 = 3 \cdot 3x \cdot 2^2 = 3 \cdot 3x \cdot 4 = 36x\)4. \(b^3 = 2^3 = 8\)

Step 3: Write the expanded polynomial

Putting it all together, the expanded form of the polynomial is: \( (3x - 2)^3 = 27x^3 - 54x^2 + 36x - 8 \).

Step 4: Simplify, if necessary

In this case, the expression is already simplified and no further simplification is needed.

Key Concepts

Polynomials

Polynomials are essential building blocks in algebra. They are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. Let’s break this down:

• A polynomial is composed of terms like \[ a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \] where each \( a_i \) represents a coefficient, \( x^i \) represents a term, and the degrees are non-negative integers.
• When you perform operations such as addition, subtraction, and multiplication with polynomials, the result is still a polynomial.

In this context, the expanded polynomial from \( (3x - 2)^3 \) is \( 27x^3 - 54x^2 + 36x - 8 \), which has several terms, each with specific powers of \( x \). This is an example of how polynomials can be transformed through algebraic operations.
Understanding polynomials is crucial as they form the basis of many algebraic concepts and real-world applications.

Cubic Binomial

A cubic binomial is a special type of polynomial that involves two terms and a degree of three. The cubic binomial format looks like \((a + b)^3\) or \((a - b)^3\). These expressions can be expanded using specific algebraic formulas.

• A binomial has exactly two terms; for example, \((3x - 2) \) represents a binomial with terms \(3x\) and \(-2\).
• When you cube a binomial, you use the formula \( (a \, \pm \, b)^3 = a^3 \, \pm \, 3a^2b \, + \, 3ab^2 \, \pm \, b^3 \), which helps in expanding the expression efficiently.

In our exercise, the interest is in the cubic binomial \((3x - 2)^3\). By applying the binomial cube formula, we derive the expanded form: \((3x - 2)^3 = 27x^3 - 54x^2 + 36x - 8\). This simplified expression helps in analyzing or solving more complex algebraic problems.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations such as addition and multiplication. They are foundational to algebra as they represent general forms of equations and can model real-world scenarios.

• For example, in the expression \( (3x - 2)^3 \), you see the use of a variable \( x \), constants \( 3 \) and \( -2 \), and the operation of exponentiation.
• These expressions allow for simplification and computational efficiency when solving algebraic problems. They can often be manipulated into equivalent expressions, just as we expanded \((3x - 2)^3\) into its expanded form.

In problem-solving, breaking down complex algebraic expressions into simpler form is imperative. It makes them easier to interpret and solve for variables, analyze functions, or apply to further algebraic studies.