Problem 118 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(b^{3}-b^{2}-2 b\)

Step-by-Step Solution

Answer

The expression factors to \(b(b - 2)(b + 1)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) of all the terms in the expression. The expression is \(b^3 - b^2 - 2b\). Notice that each term contains the variable \(b\). The GCF here is \(b\), so we factor out \(b\) from each term.

Step 2: Factor Out the GCF

Factor \(b\) out of each term in the expression: \(b^3 - b^2 - 2b = b(b^2 - b - 2)\). Now the expression is simplified to \(b(b^2 - b - 2)\).

Step 3: Factor the Quadratic Expression

Now, we focus on factoring the quadratic expression inside the parentheses: \(b^2 - b - 2\). We will look for two numbers that multiply to \(-2\) and add to \(-1\). The numbers \(-2\) and \(1\) satisfy these conditions because \(-2 \times 1 = -2\) and \(-2 + 1 = -1\).

Step 4: Express the Quadratic as a Product of Binomials

Using the numbers found, we can write the quadratic expression as a product of two binomials: \(b^2 - b - 2 = (b - 2)(b + 1)\). So the entire expression becomes \(b(b - 2)(b + 1)\).

Step 5: Verify the Factored Form

To ensure accuracy, distribute back: \(b(b - 2)(b + 1) = b[(b-2)(b+1)] = b[b^2 + b - 2b -2] = b[b^2 - b - 2] = b^3 - b^2 - 2b\). This confirms that the original expression is correctly factored as \(b(b - 2)(b + 1)\).

Key Concepts

Greatest Common Factor

When working with polynomials, especially when factoring them, one of the initial steps involves finding the Greatest Common Factor (GCF). The GCF is the largest expression that divides all terms of the polynomial without leaving any remainder. Think of it as the biggest piece that all parts of your polynomial share.
In algebraic expressions, you often focus on common variables and coefficients. For instance, in the polynomial \(b^3 - b^2 - 2b\), observe that each term contains the variable \(b\).

Here's how you determine the GCF in steps:

• Identify the lowest power of the repeated variable, which is \(b\) in our example.
• If any numerical coefficients are involved (like in terms \(3x + 6\)), find the GCF of those numbers.
• Combine these to form the GCF of the whole polynomial.

Once you have the GCF, factor it out from each term. This unclutters the expression and prepares it for further factoring if necessary.

Quadratic Expression

Quadratic expressions frequently appear in algebra. They're characterized by having a degree of 2, which means the highest power of the variable is 2. The standard form is \(ax^2 + bx + c\). Understanding their structure is crucial for efficient factoring.

In the problem \(b^2 - b - 2\), this quadratic expression needs careful handling to factor it correctly. The main goal in factoring a quadratic is to represent it as the product of two binomials. Here's a simple approach:

• Start by identifying two numbers that multiply to the constant term \(c\), here \(-2\).
• These same numbers should add up to the linear coefficient \(b\), which is \(-1\) in this expression.
• Once identified, rewrite the expression using these numbers to split the middle term, if necessary.
• Finally, factor by grouping or directly if simple, into two binomial factors.

This method relies heavily on practice and sometimes a bit of trial and error, but once you find the right pair, factoring becomes straightforward.

Binomials

Binomials are algebraic expressions that contain exactly two terms, like \(x + y\) or \(b - 2\). Understanding how to manipulate these is vital, especially when you're tasked with expanding or factoring expressions.

In the context of factoring quadratics, binomials play a key role:

• When you factor a quadratic expression, you're often expressing it as the product of two binomials. For instance, from \(b^2 - b - 2\), we derive \((b - 2)(b + 1)\).
• The coefficients and constants in these binomials correspond to the roots of the quadratic equation, or the numbers that complete the necessary sum and product in the quadratic formula.
• The process helps in simplifying calculations and solving polynomial equations by breaking complex structures into manageable pieces.

Mastering the creation and multiplication of binomials will greatly enhance your algebraic toolkit, making problems involving polynomials more accessible and less intimidating.