Factor Completely:$\[3w^3 - 27w =\\]

Feb 25, 2025 by ADMIN 37 views

Factor Completely: 3w^3 - 27w

Factoring polynomials is a crucial concept in algebra, and it's essential to understand how to factor completely. In this article, we will focus on factoring the given polynomial expression: 3w^3 - 27w. We will break down the steps involved in factoring this expression and provide a clear explanation of each step.

Before we start factoring, let's understand the given expression. The expression is a cubic polynomial, which means it has a degree of 3. The expression is 3w^3 - 27w, where w is the variable. The first step in factoring this expression is to look for any common factors.

Factoring Out the Greatest Common Factor (GCF)

The first step in factoring the expression 3w^3 - 27w is to look for the greatest common factor (GCF). The GCF is the largest factor that divides both terms of the expression. In this case, the GCF is 3w.

3w^3 - 27w = 3w(w^2 - 9)

We can factor out 3w from both terms, leaving us with 3w(w^2 - 9).

Factoring the Quadratic Expression

Now that we have factored out the GCF, we are left with a quadratic expression: w^2 - 9. This expression can be factored further using the difference of squares formula.

w^2 - 9 = (w - 3)(w + 3)

The difference of squares formula states that a^2 - b^2 = (a - b)(a + b). In this case, a = w and b = 3.

Factoring Completely

Now that we have factored the quadratic expression, we can write the factored form of the original expression.

3w^3 - 27w = 3w(w - 3)(w + 3)

This is the factored form of the original expression.

Factoring polynomials is an essential concept in algebra, and it's crucial to understand how to factor completely. In this article, we focused on factoring the expression 3w^3 - 27w. We broke down the steps involved in factoring this expression and provided a clear explanation of each step. By following these steps, you can factor any polynomial expression.

Here are some tips and tricks to help you factor polynomials:

Look for common factors: The first step in factoring any polynomial expression is to look for common factors. This can include the greatest common factor (GCF) or any other common factor that divides both terms.
Use the difference of squares formula: The difference of squares formula is a powerful tool for factoring quadratic expressions. It states that a^2 - b^2 = (a - b)(a + b).
Factor out the GCF: Factoring out the GCF is an essential step in factoring any polynomial expression. This can help you simplify the expression and make it easier to factor.

Here are some common mistakes to avoid when factoring polynomials:

Not looking for common factors: Failing to look for common factors can make it difficult to factor a polynomial expression.
Not using the difference of squares formula: The difference of squares formula is a powerful tool for factoring quadratic expressions. Failing to use it can make it difficult to factor the expression.
Not factoring out the GCF: Factoring out the GCF is an essential step in factoring any polynomial expression. Failing to do so can make it difficult to factor the expression.

Factoring polynomials has many real-world applications. Here are a few examples:

Science and Engineering: Factoring polynomials is used in science and engineering to solve equations and model real-world phenomena.
Computer Science: Factoring polynomials is used in computer science to solve problems and optimize algorithms.
Economics: Factoring polynomials is used in economics to model economic systems and solve equations.