Factor 15x3-5x2+6x-2 By Grouping. What Is The Resulting Expression

Factoring 15x³ - 5x² + 6x - 2 by Grouping: A Comprehensive Guide

Factoring polynomials is a fundamental skill in algebra, crucial for solving equations, simplifying expressions, and understanding various mathematical concepts. This article delves into the process of factoring the cubic polynomial 15x³ - 5x² + 6x - 2 using the grouping method. We'll explore the steps involved, explain the underlying principles, and provide a clear understanding of the resulting expression. Beyond this specific example, we'll also discuss the broader applicability of the grouping method and offer strategies for tackling other similar problems.

Understanding the Grouping Method

The grouping method of factoring is a technique used to factor polynomials with four or more terms. It involves grouping terms with common factors, factoring out those common factors, and then looking for further common factors between the resulting expressions. This often leads to a factored form of the original polynomial. The success of this method hinges on strategically grouping terms to reveal a common binomial factor.

Steps Involved in Factoring by Grouping

The general steps for factoring by grouping are:

1. Arrange the terms: While not always necessary, arranging the terms in descending order of powers of the variable (x in this case) can often make the grouping process clearer.

2. Group the terms: Group the terms into pairs, ensuring that each pair shares a common factor. This step requires careful observation and sometimes trial and error.

3. Factor out the greatest common factor (GCF) from each group: Identify and factor out the greatest common factor from each pair of terms.

4. Look for a common binomial factor: Examine the expressions resulting from step 3. If a common binomial factor exists, factor it out.

5. Check your work: Multiply the factored expression back out to ensure it matches the original polynomial.

Factoring 15x³ - 5x² + 6x - 2

Let's apply these steps to factor the polynomial 15x³ - 5x² + 6x - 2:

1. Arrange the terms: The terms are already arranged in descending order of powers of x.

2. Group the terms: We'll group the terms as follows: (15x³ - 5x²) + (6x - 2).

3. Factor out the GCF from each group:

   • From (15x³ - 5x²), the GCF is 5x². Factoring this out, we get 5x²(3x - 1).
   • From (6x - 2), the GCF is 2. Factoring this out, we get 2(3x - 1).

4. Look for a common binomial factor: Notice that both 5x²(3x - 1) and 2(3x - 1) share the common binomial factor (3x - 1).

5. Factor out the common binomial factor: Factoring out (3x - 1), we get (3x - 1)(5x² + 2).

Therefore, the factored form of 15x³ - 5x² + 6x - 2 is (3x - 1)(5x² + 2).

Verifying the Result

To verify our result, we can expand the factored expression:

(3x - 1)(5x² + 2) = 3x(5x² + 2) - 1(5x² + 2) = 15x³ + 6x - 5x² - 2 = 15x³ - 5x² + 6x - 2

This matches the original polynomial, confirming that our factorization is correct.

Exploring Further: Variations and Considerations

The grouping method isn't always straightforward. Sometimes, different groupings might be needed, or the method might not even work. Let's explore some variations and considerations:

Alternative Groupings

While our chosen grouping worked perfectly, it's worth noting that other groupings could be attempted. For example, you might try grouping as (15x³ + 6x) + (-5x² - 2). While this grouping might seem less obvious, it might still lead to the same factored form, though the process could be slightly more complex. Experimentation is key.

Polynomials Where Grouping Doesn't Work

Not all polynomials with four terms are factorable by grouping. Sometimes, the polynomial might be prime (cannot be factored into simpler expressions with integer coefficients). In these cases, other factorization techniques might be necessary, or the polynomial may not factor nicely at all.

Advanced Applications of Grouping

The grouping method isn't limited to cubic polynomials. It can be applied to polynomials of higher degrees with the appropriate number of terms and strategic groupings. The principle remains the same: find common factors within groups and then identify common binomial (or even trinomial) factors.

Connection to Other Factoring Techniques

The grouping method is often used in conjunction with other factoring techniques, such as factoring out a greatest common factor (GCF) before attempting grouping. For instance, if you encounter a polynomial with a common factor across all terms, factoring out that GCF first simplifies the expression and often makes the grouping process easier.

For example, consider the polynomial 30x³ - 10x² + 12x - 4. Notice that all terms have a common factor of 2. Factoring out 2, we get 2(15x³ - 5x² + 6x - 2). We've already factored 15x³ - 5x² + 6x - 2, so the fully factored form is 2(3x - 1)(5x² + 2).

Practical Applications and Problem-Solving Strategies

Factoring polynomials, including the grouping method, is a fundamental skill with numerous applications across various branches of mathematics and other fields. These include:

• Solving polynomial equations: Factoring a polynomial allows us to find its roots (or zeros), which represent the solutions to the corresponding polynomial equation.

• Simplifying algebraic expressions: Factoring can significantly simplify complex algebraic expressions, making them easier to manipulate and understand.

• Calculus: Factoring is essential in calculus, especially in finding derivatives and integrals.

• Engineering and Physics: Polynomials are used extensively in modeling various physical phenomena. Factoring helps in analyzing and understanding these models.

Tips for Successful Factoring by Grouping

• Practice: The best way to master factoring by grouping is through consistent practice. Work through numerous examples to build your intuition and recognize patterns.

• Systematically try different groupings: If one grouping doesn't work, try another. There's often more than one way to group the terms.

• Check your work carefully: Always multiply the factored expression back out to verify that it matches the original polynomial.

• Consider other factoring techniques: If grouping doesn't lead to a solution, explore other techniques like factoring trinomials or using the quadratic formula if applicable.

Conclusion

Factoring the polynomial 15x³ - 5x² + 6x - 2 by grouping demonstrates a powerful technique for simplifying algebraic expressions. The resulting expression, (3x - 1)(5x² + 2), highlights the effectiveness of this method. By understanding the steps involved and exploring the variations and considerations discussed, you can effectively apply the grouping method to a wide range of polynomial factoring problems, enhancing your algebraic skills and opening doors to more advanced mathematical concepts. Remember that consistent practice is key to mastering this valuable tool.