Problem 58 - Beginning and Intermediate Algebra

Question

Factor by grouping. \(3 j k-7 k+6 j-14\)

Step-by-Step Solution

Answer

The factored form of the given expression by grouping is \((3j - 7)(k + 2)\).

Step 1: Group terms with common factors

We will group the terms with common factors: \((3jk - 7k) + (6j - 14)\)

Step 2: Factor out the common factors

Now, we will factor out the common factors from each group: \(k(3j - 7) + 2(3j - 7)\)

Step 3: Factor out the common expression

We observe that both terms have a common expression \((3j-7)\), so we factor it out: \((3j - 7)(k + 2)\) In conclusion, the factored form of the given expression by grouping is \((3j - 7)(k + 2)\).

Key Concepts

Understanding Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients that involve the operations of addition, subtraction, multiplication, and non-negative integer exponents. In simpler terms, they are combinations of several terms. Each term is made up of a constant, known as the coefficient, and one or more variables raised to whole number powers.
Polynomials appear frequently in algebra and are the building blocks for understanding complex equations in calculus and beyond. In our exercise, the expression is a polynomial with four terms: \(3jk\), \(- 7k\), \(6j\), and \(-14\). These terms have different combinations of variables \(j\) and \(k\), each with a whole number exponent (or implicitly 1 when no exponent is shown).

• Terms: Parts of the polynomial separated by signs like '+' or '-'.
• Degree: The highest power of the variable in the polynomial, determined by adding together the powers of the individual variables within a term.
• Variable: A symbol for a number we do not know yet; typically \(j\) and \(k\) in our example.

Understanding the structure of polynomials is essential as it sets the stage for different methods of simplifying and solving these expressions.

Mastering Factor by Grouping

Factor by grouping is a technique used to simplify polynomials by rearranging and combining terms to reveal common factors. This method is particularly useful when dealing with four-term polynomials, like the exercise provided.
The process begins by dividing the polynomial into two pairs and factoring out the greatest common factor from each group.
On the original polynomial \(3jk - 7k + 6j - 14\), we first group them as follows:

• Group 1: \( (3jk - 7k) \)
• Group 2: \( (6j - 14) \)

Within each group, we identify a common factor:

• Group 1: Factor out \(k\), leaving \(k(3j - 7)\).
• Group 2: Factor out 2, resulting in \(2(3j - 7)\).

Both of these groups now share a common expression, \((3j - 7)\), which can be factored out, giving us the factored form \( (3j - 7)(k + 2) \).
By pulling out these common factors, we simplify the polynomial significantly, making it easier to work with in algebraic equations.

Working with Mathematical Expressions

Mathematical expressions are combinations of numbers, symbols, and operators (like '+' and '-') that represent a specific value or computational process. In algebra, expressions like \(3jk - 7k + 6j - 14\) are common. They don’t always need to be solved but can be simplified or rewritten to find equivalences or derive further insights.
Simplifying expressions often involves finding like terms, identifying common factors, or using methods like factor by grouping. Grouping helps to identify patterns or structures in expressions that make it easier to simplify or manipulate them.

• Like terms: Terms that have the same variables raised to the same powers. For instance, \(3jk\) and \(6j\) cannot be combined because their variables differ.
• Rewriting expressions: Altering the form of an expression to make it easier to understand or work with, without changing its value.

By learning to manage these expressions, students become adept at making calculations more straightforward and solving complex problems in algebra and beyond.