Problem 19 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$3 d^{2}-6 d$$

Step-by-Step Solution

Answer

The factored form of the given expression is: \(3d(d - 2)\).

Step 1: Identify the coefficients and variables

In the given expression, \(3d^2 - 6d\), the coefficients are 3 in front of the \(d^2\) and -6 in front of the d term. The variable is d.

Step 2: Find the GCF of coefficients

The coefficients are 3 and -6. The greatest common factor of the numbers 3 and 6 is 3. So, GCF of the coefficients is 3.

Step 3: Find the GCF of variables

The variables are d (first term) and d (second term). The GCF of these variables is the smallest power of d, which is d.

Step 4: Combine the GCF of coefficients and variables

Now, we need to combine the GCF of the coefficients and variables to find the overall GCF of the expression. GCF of coefficients = 3 GCF of variables = d Overall GCF of the expression = 3d

Step 5: Factor out the GCF from the given expression

Now, factor out the overall GCF (3d) from the given expression \(3d^2 - 6d\). When we factor out 3d, we obtain the following expression: \(3d(d - 2)\)

Step 6: Check the answer

To verify our answer, let's expand the factored form of the expression: \(3d(d - 2) = 3d^2 - 6d\) As the expanded form matches the original expression, our answer is correct. The factored form of the given expression is: \(3d(d - 2)\)

Key Concepts

Greatest Common Factor

One of the foundational concepts in algebra is finding the Greatest Common Factor (GCF), which is crucial in simplifying algebraic expressions. The GCF of a set of terms is the largest factor that can divide each term evenly without leaving a remainder. When determining the GCF:

• Look at the numerical coefficients (the numbers in front of variables). For example, in the expression \(3d^2 - 6d\), the coefficients are 3 and -6.
• Find the GCF of these numbers—here, it is 3 because 3 is the largest number that divides both 3 and 6 evenly.
• Look at the variables and select the one with the smallest power from all terms.

Finding the GCF helps us simplify expressions by factoring them. Factoring larger expressions becomes easier when we break them down into smaller parts.

Polynomial Factorization

Polynomial factorization is a process of expressing a polynomial as a product of its factors. Factoring simplifies expressions and allows us to solve for variables more easily.Consider the expression \(3d^2 - 6d\). By identifying the GCF, 3d, we transform it into a simpler form, \(3d(d - 2)\).Here’s the step-by-step approach:

• Identify the common factors in all terms, both numerical and variable-based.
• Write the original polynomial as a product of the GCF and the remaining polynomial terms.
• Use distribution to verify the factorization is correct.

Polynomial factorization is essential in algebra for solving equations and simplifying expressions to make calculations more manageable.

Algebraic Expressions

Algebraic expressions consist of terms made up of coefficients and variables. It's crucial to understand how to manipulate these expressions for problem-solving in algebra.For example, in \(3d^2 - 6d\):

• The algebraic expression comprises two terms: \(3d^2\) and \(-6d\).
• Each term has a coefficient (3 and -6) and a variable part \(d\).
• Expressions can be simplified by factoring, combining like terms, or expanding.

Mastery in handling algebraic expressions lays the groundwork for learning other math topics, like equations and functions.

Basic Algebra Concepts

Understanding basic algebra concepts serves as a building block for advanced mathematical learning. These concepts include arithmetic operations, variable manipulation, and understanding expressions. Key elements include:

• Arithmetic operations including addition, subtraction, multiplication, and division.
• Concepts like GCF, LCM, and variable understanding.
• Simplification methods like factoring and expanding expressions.

When students grasp these basic concepts, they can tackle more complex problems, as algebra is an important aspect of higher mathematics.