Problem 114 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(54-16 x^{3}\)

Step-by-Step Solution

Answer

The completely factored form is: \(2(3 - 2x)(9 + 6x + 4x^2)\).

Step 1: Look for Common Factors

First, identify any common factors present in the terms of the expression. We have two terms: 54 and \(-16x^3\). The greatest common factor between 54 and 16 is 2. So, we can factor 2 out of the expression:\[54 - 16x^3 = 2(27 - 8x^3)\]

Step 2: Recognize Special Forms

The expression inside the parentheses, \(27 - 8x^3\), is a difference of cubes. Recall that a difference of cubes \(a^3 - b^3\) can be factored into \((a - b)(a^2 + ab + b^2)\). Here, \(27 = 3^3\) and \(8x^3 = (2x)^3\). Thus, the expression can be written as a difference of cubes \((3^3 - (2x)^3)\).

Step 3: Apply the Difference of Cubes Formula

Apply the difference of cubes formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), where \(a = 3\) and \(b = 2x\). Substitute these values into the formula:\[(3 - 2x)((3)^2 + 3(2x) + (2x)^2)\]Simplify the terms:\[(3 - 2x)(9 + 6x + 4x^2)\]

Step 4: Write the Completely Factored Form

Now, combine the factored expressions from steps 1 and 3 to get the completely factored form:\[2((3 - 2x)(9 + 6x + 4x^2))\]This is the fully factored expression of \(54 - 16x^3\).

Key Concepts

Difference of Cubes

When we encounter a math expression like the one inside the parentheses, we can recognize it as a difference of cubes. This means our expression is in the form of \(a^3 - b^3\). To factor these, we can use a handy formula:

• For \(a^3 - b^3\), the formula is \((a - b)(a^2 + ab + b^2)\).

Now, let’s see how the formula works with our example:The number 27 is a cube number because it is equal to \(3^3\).Also, \(8x^3\) is another cube, as it is equal to \((2x)^3\).Replacing \(a\) with 3 and \(b\) with \(2x\):- Begin with \(3 - 2x\);- Then, go on to \((3)^2 + 3(2x) + (2x)^2\).This results in our factored form: \((3 - 2x)(9 + 6x + 4x^2)\).
Difference of cubes is a neat trick to simplify our expressions in algebra! Remembering this formula can make solving similar problems a breeze.

Common Factors

Start by finding any shared factors in an expression's terms. This first step is called identifying common factors. Here’s why it’s important:

• Finding a common factor simplifies our work by shrinking the original problem into a more manageable form.

Take the expression, \(54 - 16x^3\).Notice the terms:

• 54, and
• -16, the number associated with \(x^3\).

The greatest common factor between these two numbers is 2. By factoring out the 2, we reduce our expression to:\[54 - 16x^3 = 2(27 - 8x^3)\],prepping it for more factoring down the line.
This makes identifying common factors a valuable tool in algebra.

Algebraic Expressions

Algebraic expressions may seem intimidating, but they are simply combinations of numbers, variables, and operations.These expressions can appear on their own or within equations.For example, \(54 - 16x^3\) mixes numbers with the variable \(x\).

• The task is often to simplify or solve them by factoring.
• Consider expressions as puzzles that we break down into smaller pieces.

By recognizing patterns like the difference of cubes or identifying common factors, we transform complex expressions into straightforward tasks.Learning these methods not only improves problem-solving skills but also enhances understanding of different relationships in math.
Embrace algebraic expressions, as they are the building blocks of many math concepts!