Problem 53 - Beginning and Intermediate Algebra

Question

Factor by grouping. $$k t+3 k+8 t+24$$

Step-by-Step Solution

Answer

The simplified expression after factoring by grouping is \((t + 3)(k + 8)\).

Step 1: Group the terms

First, we need to group the terms that contain common factors. In our given expression, we can group the first two terms together and the last two terms together as follows: \((k t + 3 k) + (8 t + 24)\)

Step 2: Factor each group

Now, let's find the common factors of the grouped terms and factor them by dividing each term by the common factor. \(k t + 3 k\) has a common factor of \(k\). So, let's take out \(k\) as a common factor: \(k (t + 3)\) \(8 t + 24\) has a common factor of \(8\). So, let's take out \(8\) as a common factor: \(8 (t + 3)\) Now, our expression looks like this: \(k (t + 3) + 8 (t + 3)\)

Step 3: Factor the common term

Finally, we can see that \((t + 3)\) is a common term in both parts of the expression. Let's factor it out: \((t + 3) (k + 8)\) Our factored expression is: \((t + 3)(k + 8)\)

Key Concepts

Common Factors

In algebra, a common factor is a number or a variable that divides two or more algebraic terms without leaving a remainder. Recognizing common factors is essential when dealing with algebraic expressions, as it allows us to simplify the expression by pulling out the greatest common factor.

For example, in the expression \(kt + 3k\), both terms share a common factor of \(k\). Similarly, in the expression \(8t + 24\), the common factor is \(8\).

• The 'common factor' allows us to rewrite the expression in a more manageable form.
• It simplifies the polynomial and makes further manipulation possible.
• Identifying common factors is a crucial step in the process of factoring by grouping.

By factoring out the common factors, you simplify the expression and set the stage for further factoring. This approach not only organizes your work but also lays the foundation for solving more complex algebraic equations later on.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can represent a single value or a range of values, depending on the variables involved. Understanding how to manipulate these expressions is foundational in algebra.

In the context of factoring by grouping, we deal with expressions such as \(kt + 3k + 8t + 24\). This expression consists of four terms, each involving coefficients and variables.

• Each term in an algebraic expression can potentially have different properties that can be identified and manipulated.
• The ability to rearrange and group these terms is key for processes such as factoring by grouping.
• The structure of an expression influences how we approach techniques that simplify or solve it.

Factoring by grouping becomes a valuable tool when we encounter expressions with four or more terms. The goal is to enhance the expression's form, making it easier to interpret or solve further.

Factoring Polynomials

Factoring polynomials is the mathematical process of breaking down a polynomial into simpler 'factor' polynomials that can be multiplied together to give the original polynomial back. This is important in solving polynomial equations, simplifying expressions, and finding polynomial roots.

In factoring by grouping, we strategically group terms within a polynomial to reveal common factors. In the example \(kt + 3k + 8t + 24\), we initially group it as \((kt + 3k) + (8t + 24)\). Next, we factor out the common factors from each group: \(k(t + 3)\) and \(8(t + 3)\). Finally, noticing that \(t + 3\) appears in both, we can factor it out, resulting in \((t + 3)(k + 8)\).

• Grouping helps identify patterns or structures within a polynomial that might not be immediately apparent.
• By systematically factoring each group, you ensure precision and avoid errors.
• Recognizing common factors in grouped terms facilitates the final factoring step, resulting in the simplified product of two binomials.

Factoring polynomials is a vital algebraic skill, enabling the solution of equations and the simplification of complicated expressions. Mastery of this technique offers enhanced confidence and problem-solving ability in more advanced mathematics.