Completely Factoring 10xy + 35x + 6y + 21
This article details the complete factorization of the algebraic expression 10xy + 35x + 6y + 21, exploring various factoring techniques and demonstrating the verification process. We will examine factoring by grouping, the greatest common factor (GCF) method, and discuss the applicability of other methods. The goal is to provide a comprehensive understanding of the steps involved and the rationale behind each choice.
Understanding the Expression
The expression 10xy + 35x + 6y + 21 consists of four terms: 10xy, 35x, 6y, and 21. Factoring in algebra involves rewriting an expression as a product of simpler expressions. Several techniques exist, each suited to different types of expressions. These include factoring by grouping, finding the greatest common factor (GCF), difference of squares, and trinomial factoring.
Factoring by Grouping
Factoring by grouping is a technique used when an expression has four or more terms. It involves grouping terms with common factors and then factoring out those common factors. For 10xy + 35x + 6y + 21:
- Group the terms: (10xy + 35x) + (6y + 21)
- Factor out the GCF from each group: 5x(2y + 7) + 3(2y + 7)
- Factor out the common binomial factor: (2y + 7)(5x + 3)
Therefore, the completely factored form is (2y + 7)(5x + 3).
Example: Consider the expression 12ab + 8a + 15b + 10. Grouping gives (12ab + 8a) + (15b + 10), which factors to 4a(3b + 2) + 5(3b + 2) and finally (3b + 2)(4a + 5).
| Method | Steps | Example | Advantages/Disadvantages |
|---|---|---|---|
| Factoring by Grouping | Group terms, factor out GCF from each group, factor out common binomial. | 10xy + 35x + 6y + 21 = (2y + 7)(5x + 3) | Effective for expressions with four or more terms; can be less efficient if no common binomial factor exists. |
| GCF | Find the greatest common factor of all terms and factor it out. | No single GCF for all terms in this expression. | Simple and efficient when a common factor exists; ineffective if no common factor is present. |
| Difference of Squares | Applicable only to expressions in the form a² – b². | Not applicable to this expression. | Very efficient for applicable expressions; limited to specific forms. |
| Trinomial Factoring | Applicable to trinomials of the form ax² + bx + c. | Not applicable to this expression. | Effective for trinomials; not applicable to expressions with more than three terms. |
Greatest Common Factor (GCF)
While there’s no single GCF for all four terms in 10xy + 35x + 6y + 21, identifying potential GCFs within groups is crucial for factoring by grouping, as demonstrated above. In this case, the GCF of 10xy and 35x is 5x, and the GCF of 6y and 21 is 3. Factoring out these GCFs within their respective groups simplifies the expression and allows for the identification of the common binomial factor (2y + 7).
Alternative Factoring Methods
Difference of squares and trinomial factoring are not directly applicable to this expression because it has four terms and doesn’t fit the specific forms required for these methods. Factoring by grouping proves to be the most efficient method in this case.
- Factoring by Grouping: Strengths: Effective for expressions with four or more terms. Weakness: Requires a common binomial factor.
- GCF: Strengths: Simple and efficient when a common factor exists. Weakness: Ineffective if no common factor is present for all terms.
- Difference of Squares: Strengths: Efficient for expressions in the form a² – b². Weakness: Limited applicability.
- Trinomial Factoring: Strengths: Effective for trinomials. Weakness: Not applicable to expressions with more than three terms.
Factoring by grouping is more efficient than other methods when dealing with expressions containing four or more terms that can be grouped to reveal a common binomial factor, as seen in this example.
Verification of the Factored Expression
To verify the factored expression (2y + 7)(5x + 3), we expand it using the distributive property (FOIL method):
- First: (2y)(5x) = 10xy
- Outer: (2y)(3) = 6y
- Inner: (7)(5x) = 35x
- Last: (7)(3) = 21
Combining these terms gives 10xy + 6y + 35x + 21, which is the original expression. This confirms that (2y + 7)(5x + 3) is the correct factorization.
A visual representation of the verification would show the original expression at the top, then the steps of expanding the factored form (using arrows to connect terms and show the multiplication process), culminating in the original expression at the bottom, demonstrating the equivalence.
