Question

Factor out the common factor.\(3 x^{3}-6 x\)

Short answer

The common factor of \(3 x^{3}-6 x\) is \(3x\), so factoring this common factor out gives \(3x * (x^{2} - 2)\).

Step-by-step solution

Identify the Common Factor

Examining the monomials \(3x^{3}\) and \(-6x\), the common factor is \(3x\). Because both of these terms are divisible by \(3x\).

Factor Out the Common Factor

Dividing both terms by the common factor, \(3x\), provides the remaining terms of the expression when factored out. Hence, \(\frac{3x^{3}}{3x} = x^{2}\) and \(\frac{-6x}{3x} = -2\). Therefore, placing these within brackets the factored form of the expression is \(3x * (x^{2} - 2)\).

Verification

To verify if the expression has been factored correctly, expand the factored form \(3x * (x^{2} - 2)\) and check if the result is equal to the original expression \(3x^{3} - 6x\). Doing this shows that the factored equation is correct because the expanded expression equals the original one.

Key concepts

Understanding Common Factors

When factoring polynomials, one of the key steps is to find the common factors in the expression. A common factor is a term, or terms, that are present in each component of the polynomial. For example, in the expression \(3x^3 - 6x\), both terms share \(3x\) as a common factor. This means that \(3x\) evenly divides each term in the polynomial.

The process of identifying common factors involves:

• Looking at each monomial (individual term) in the polynomial.
• Finding coefficients and variables that appear in each term with the same power.
• Ensuring the highest power of each common variable is included.

By factoring out these common elements, complex expressions can be simplified considerably, paving the way for further algebraic manipulation or solving.

Breaking Down Algebraic Expressions

An algebraic expression consists of variables, constants, and operations that form a meaningful mathematical phrase. Factoring an expression like \(3x^3 - 6x\) involves breaking it down into simpler parts. This breakdown is crucial for solving equations, simplifying calculations, or finding roots.

When handling algebraic expressions, it's often helpful to:

• Identify terms (individual elements of the expression separated by plus or minus signs).
• Note the coefficient (number in front of the variable) and the variables (e.g., \(x\) in \(3x^3\)).
• Ensure proper mathematical rules are followed during the factoring process.

This makes handling algebraic expressions not only manageable but also more approachable, even for complex calculations.

Verification of Solutions in Factoring

Once you've factored a polynomial, it's important to verify your solution. This ensures that the factorization process was done correctly. Verification involves expanding the factored expression and checking if it matches the original polynomial.

In the case of our example, the verified factored form \(3x \times (x^2 - 2)\) is expanded to reveal:\[3x(x^2) - 3x(2) = 3x^3 - 6x\]This step confirms the correctness of the factorization as the expression equals the original polynomial. Verification not only provides reassurance but also reinforces understanding of algebraic principles.

Whenever working with polynomials, implementing a quick verification step can save time and avoid errors in more complicated mathematical tasks.