Problem 43 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$2 k^{2}(3 r+4)-(3 r+4)$$

Step-by-Step Solution

Answer

\( (3r+4)(2k^2-1) \)

Step 1: Identify the Greatest Common Factor (GCF)

First, let's identify the greatest common factor between the two expressions $$2k^2(3r+4)$$ and $$(3r+4)$$. We can see that both expressions have a common factor of $$(3r+4)$$.

Step 2: Factor out the GCF

Now, let's factor out the greatest common factor $$(3r+4)$$ from the initial expression: $$(3r+4)(2k^2-1)$$

Step 3: Simplification and Checking the Answer

Now that we have the factored expression, there is no need for further simplification as there is no other common factor. Let's check our answer by expanding the factored expression: $$(3r+4)(2k^2-1) = (3r+4) \cdot 2k^2 - (3r+4) \cdot 1 = 2k^2(3r+4) - (3r+4)$$ Our factored expression is correct, and the final answer is: $$(3r+4)(2k^2-1)$$

Key Concepts

The Greatest Common Factor (GCF)

The concept of the greatest common factor, often abbreviated as GCF, plays a pivotal role in the process of factoring polynomials. The GCF of two or more polynomials is the largest polynomial that divides each of the polynomials without leaving a remainder.
In our example, we can see that the polynomial \(2k^2(3r+4)\) and \((3r+4)\) share a common factor of \((3r+4)\). Recognizing the GCF makes the factoring process easier as it allows us to "pull out" or "factor out" this common term, simplifying the expression. Here’s how you can identify the GCF:

• Look for common numerical coefficients, if present.
• Identify any shared variables or polynomials.
• Select the highest power of these shared elements.

Understanding the GCF is the foundation for polynomial factorization, and ensures that you can break down complex expressions into simpler, more manageable parts.

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication). These expressions can represent real-world quantities, abstract concepts, or complex mathematical ideas. In essence, they are the building blocks of algebra.When dealing with expressions, especially for factoring exercises, it's important to remember:

• Variables like \(k, r\) act as placeholders for numbers and can have coefficients, like \(2k^2\).
• Terms are individual parts of an expression, separated by addition or subtraction, such as \(2k^2\) and \(3r+4\).
• Operations within expressions follow specific rules, such as the distributive property and combining like terms.

By thoroughly understanding algebraic expressions, you are better prepared to factor them effectively, identify the greatest common factors, and manipulate polynomials with ease.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of simpler polynomials. This process simplifies expressions and is crucial for solving polynomial equations and analyzing their behavior.In our specific exercise, we factored the polynomial \(2k^2(3r+4)-(3r+4)\) by:

• Identifying the shared term \((3r+4)\) as the greatest common factor.
• Factoring out \((3r+4)\), which leaves us with: \((3r+4)(2k^2-1)\).
• Verifying our factorization by expanding to check if we return to the original expression.

Knowing how to factor a polynomial is essential for algebra, as it allows you to solve equations efficiently and understand the properties of polynomial functions. Mastering this skill opens the door to advanced algebraic techniques and deeper mathematical understanding.