Problem 93 - College Algebra with Modeling and Visualization

Question

Multiply the expressions. $$3 x(x+1)(x-1)$$

Step-by-Step Solution

Answer

The expression simplifies to \( 3x^4 - 3x^2 \).

Step 1: Distribute the First Term

Start by multiplying the first two factors in the expression: \( x(x + 1) = x^2 + x \).This simplifies the expression to:\( 3x(x^2 + x)(x-1) \).

Step 2: Distribute the Second Term

Next, multiply the result from Step 1 with the third factor:\( (x^2 + x)(x - 1) = x^2(x - 1) + x(x - 1) \).This requires distributing as follows:\[ x^2(x - 1) = x^3 - x^2 \]\[ x(x - 1) = x^2 - x \]So, combining these results we have:\( x^3 - x^2 + x^2 - x \), which simplifies to:\( x^3 - x \).

Step 3: Final Distribution and Simplification

Finally, multiply this result by the original leading constant term:\( 3x(x^3 - x) \).Distributing gives:\[ 3x imes x^3 = 3x^4 \]\[ 3x imes (-x) = -3x^2 \].So, the entire expression simplifies to:\( 3x^4 - 3x^2 \).

Key Concepts

Expression Multiplication

When working with polynomials and algebra, expression multiplication is one of the fundamental operations. It involves multiplying terms in expressions to combine them into a single statement. This process is essential as it allows us to easily expand polynomials or simplify complex expressions. To tackle expression multiplication, we start by multiplying individual terms or factors one at a time in a methodical way. For example, given three expressions like in the exercise, we start by targeting two expressions at a time.

• Identify each expression or component within the problem.
• Carefully multiply them together using distribution.
• Proceed with combining any like terms resulting from the multiplication.

In our specific problem, the expression is multiplied in steps. First, two expressions are multiplied to get a resultant expression, and it is further multiplied with the next component. It's about building the expression step by step, with patience and attention to detail. As we follow the order of multiplication, taking care of brackets and distribution helps manage the multiplication smoothly.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form while maintaining the same value. This process unclutters algebra and makes working with equations easier and more intuitive. In the given problem, it is crucial to simplify expressions as early as possible after each multiplication step. This keeps the expression manageable and helps in spotting further simplifications quickly.

• After multiplying expressions, look for like terms. These are terms that have the same variable raised to the same power.
• Add or subtract like terms to simplify the expression.
• Keep simplifying after each multiplication or distribution step as early as possible for best clarity.

Using these strategies, expressions become streamlined, concise, and ready for further operations, solving, or analysis. In our solution, the expression was successively simplified at each stage by combining like terms and removing any zero-sum parts such as \(x^2 - x^2\), which simplifies to zero.

Polynomial Expansion

Polynomial expansion happens when expressions involving polynomials are multiplied, and the result is expressed in standard polynomial form. This expansion is a customary part of algebra that allows us to deal with polynomials in an extended format. In the exercise solution, polynomial expansion was handled step by step, solidifying the idea that each phase should be tackled with care.

• Begin with breaking down complex multiplications into simpler parts.
• Apply the distributive property multiple times across different groups of terms.
• Keep expanding until each term feels atomic or indivisible.

In our case, the polynomial \(3x(x+1)(x-1)\) was expanded by initially distributing among internal terms before involving more complex factors like the leading coefficient. Through methodical application of distribution, the expanded polynomial \(3x^4 - 3x^2\) was achieved efficiently.