Completely Factoring Algebraic Expressions
This article details the process of completely factoring the algebraic expression 10xy + 35x + 6y + 21. We will explore various factoring techniques, including factoring by grouping and the use of the greatest common factor (GCF), to arrive at the completely factored form. The process will be explained step-by-step, along with verification methods to ensure accuracy.
Understanding the Expression
The expression 10xy + 35x + 6y + 21 contains four terms: 10xy, 35x, 6y, and 21. Each term consists of a coefficient (a numerical factor) and one or more variables. Factoring an algebraic expression involves rewriting it as a product of simpler expressions. Expressions with four terms are often factored using the method of grouping.
Factoring by Grouping
Factoring by grouping involves separating the expression into groups of terms, factoring out the greatest common factor from each group, and then factoring out a common binomial factor. This method is particularly useful for expressions with four terms.
Examples:
- 2ab + 2ac + 3b + 3c = 2a(b + c) + 3(b + c) = (2a + 3)(b + c)
- x³ + x² + 4x + 4 = x²(x + 1) + 4(x + 1) = (x² + 4)(x + 1)
- 6xy – 9x + 4y – 6 = 3x(2y – 3) + 2(2y – 3) = (3x + 2)(2y – 3)
Choosing appropriate groups often involves looking for common factors among pairs of terms. Sometimes, experimentation is necessary to find the most effective grouping.
| Method | Description | Advantages | Disadvantages |
|---|---|---|---|
| Factoring by Grouping | Separating the expression into groups and factoring out common factors from each group. | Effective for expressions with four or more terms. | May require trial and error to find the correct grouping. |
| Greatest Common Factor (GCF) | Finding the largest factor common to all terms and factoring it out. | Simplifies the expression. | Not always applicable to all expressions. |
| Difference of Squares | Factoring expressions of the form a² – b² as (a + b)(a – b). | Simple and efficient for specific expressions. | Only applicable to expressions in the form a² – b². |
| Trinomial Factoring | Factoring expressions of the form ax² + bx + c. | Widely applicable to quadratic expressions. | Can be more complex than other methods. |
Greatest Common Factor (GCF)
The greatest common factor (GCF) of the terms in the expression 10xy + 35x + 6y + 21 is 1. While there is no common factor among all four terms, identifying the GCF of subsets of terms is crucial for factoring by grouping.
In this case, the GCF helps us to initially identify potential groupings for the factoring by grouping method. Although the overall GCF is 1, examining the GCF of pairs of terms guides the grouping process.
Flowchart for Finding and Using the GCF:
1. Identify all terms in the expression.
2. Find the prime factorization of each term’s coefficient and variables.
3. Identify common factors across all terms.
4. Multiply the common factors to find the GCF.
5. Factor out the GCF from the expression.
6. Check if the remaining expression can be factored further.
Verification of the Factored Form
To verify a factored form, expand the factored expression using the distributive property (FOIL method if applicable). The result should match the original expression.
- Expand the factored expression.
- Combine like terms.
- Compare the simplified expression to the original expression.
- If they are identical, the factored form is correct.
Example of an Incorrect Factored Form: If, incorrectly, we were to factor 2x² + 5x + 3 as (2x + 1)(x + 3), expanding this gives 2x² + 7x + 3, which is not the original expression. The correct factorization is (2x + 3)(x + 1).
Alternative Factoring Methods
For this specific expression (10xy + 35x + 6y + 21), factoring by grouping is the most efficient method. Alternative methods, such as trial and error for factoring trinomials, are not directly applicable because it’s not a trinomial.
Illustrative Example with Detailed Steps
Let’s factor 10xy + 35x + 6y + 21 completely using factoring by grouping:
- Group terms: (10xy + 35x) + (6y + 21)
- Factor out GCF from each group: 5x(2y + 7) + 3(2y + 7)
- Factor out the common binomial: (5x + 3)(2y + 7)
Verification: Expanding (5x + 3)(2y + 7) using the FOIL method yields 10xy + 35x + 6y + 21, confirming the factored form is correct.
