Factoring 30x² + 40xy + 51y² A Step By Step Analysis

Factoring polynomials is a fundamental skill in algebra, enabling us to simplify complex expressions and solve equations more efficiently. When faced with a polynomial like 30x² + 40xy + 51y², it's crucial to determine the most accurate way to factor it. This article provides a detailed exploration of the factorization of this polynomial, assessing different options and clarifying the correct approach. We'll examine the given statements to identify which one accurately represents the factorization process.

Understanding the Polynomial

Before diving into specific factorization methods, let's understand the structure of the polynomial 30x² + 40xy + 51y². This is a quadratic polynomial in two variables, x and y. The terms include a squared term in x (30x²), a squared term in y (51y²), and a mixed term (40xy). To factor this polynomial effectively, we need to consider various techniques, including:

• Greatest Common Factor (GCF): Identifying the largest factor common to all terms.
• Trial and Error: Attempting to find two binomials that multiply to give the original polynomial.
• Completing the Square: Transforming the polynomial into a perfect square trinomial plus a constant term.
• Quadratic Formula: Using the quadratic formula to find the roots and then construct the factors.

The key to successful factorization lies in selecting the appropriate method based on the polynomial's characteristics. In this case, we will meticulously go through each proposed statement to determine the correct factorization.

Evaluating Statement A: Factoring out 10

Statement A suggests that the polynomial can be rewritten after factoring as 10(3x² + 4xy + 5y²). To verify this, we need to distribute the 10 back into the expression and see if it matches the original polynomial.

• Distributing 10: 10 * (3x² + 4xy + 5y²) = 30x² + 40xy + 50y²

Comparing this result with the original polynomial 30x² + 40xy + 51y², we observe a discrepancy in the last term. The original polynomial has 51y², while the factored expression yields 50y². Therefore, factoring out 10 as suggested in statement A does not accurately represent the factorization of the given polynomial. This indicates that statement A is incorrect.

Detailed Analysis of GCF

The greatest common factor (GCF) is a crucial concept in polynomial factorization. It involves finding the largest factor that divides all terms in the polynomial. In the given polynomial, 30x² + 40xy + 51y², we examine the coefficients (30, 40, and 51) and the variables (x² , xy, and y²).

• Coefficients: The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. The factors of 51 are 1, 3, 17, and 51. The greatest common factor of 30, 40, and 51 is 1.
• Variables: The terms have x² , xy, and y². There is no common variable factor among all three terms.

Since the GCF of the coefficients is 1 and there are no common variable factors, the greatest common factor for the entire polynomial is 1. This means we cannot simplify the polynomial by factoring out a common factor other than 1. Statement A's attempt to factor out 10 is therefore incorrect because 10 is not a common factor of all the coefficients, particularly 51.

In summary, a meticulous analysis of the greatest common factor reveals that statement A is not a valid factorization of the given polynomial. The presence of 51 as a coefficient, which does not share a factor of 10, confirms this conclusion. Therefore, it is essential to consider other factorization methods to accurately represent the given polynomial.

Examining Statement B: Product of a Trinomial and xy

Statement B proposes that the polynomial can be rewritten as the product of a trinomial and xy. This suggests a factorization pattern where 30x² + 40xy + 51y² = (ax² + bxy + cy²)(xy) for some constants a, b, and c. However, this form of factorization is highly unlikely because distributing xy would result in terms with powers of x and y that are higher than what we have in the original polynomial.

Why Statement B is Unlikely

To understand why this is improbable, let's consider the degrees of the terms that would result from such a factorization:

• (ax²)(xy) = ax³y: This term has a degree of 4 (3 in x and 1 in y), which is not present in the original polynomial.
• (bxy)(xy) = bx²y²: This term also has a degree of 4, which is not in the original polynomial.
• (cy²)(xy) = cxy³: This term has a degree of 4 as well, further confirming that this form of factorization would introduce terms of higher degrees than those in the original polynomial.

Since the original polynomial 30x² + 40xy + 51y² only contains terms of degree 2, factoring it into a product involving xy and a trinomial would not yield the correct expression. The degrees of the terms after distribution would be inconsistent with the original polynomial.

Testing a Hypothetical Factorization

Let's illustrate this with a hypothetical example. Suppose we assume that the factorization takes the form:

30x² + 40xy + 51y² = (ax + by + c)(xy)

Expanding the right side, we get:

axy² + bxy² + cxy

This expansion does not resemble the original polynomial. We would expect terms like x² and y², which are clearly missing in this expanded form. The presence of terms with degree 3 (such as x²y or xy²) is also problematic, as the original polynomial only has terms of degree 2.

Therefore, statement B is incorrect because factoring the polynomial as a product of a trinomial and xy introduces terms with higher degrees than those present in the original polynomial, and the resulting expression would not be equivalent to 30x² + 40xy + 51y². This analysis reinforces the need to explore other potential factorization methods that align with the polynomial's structure and degree.

Conclusion: Identifying the Correct Statement

After thoroughly evaluating both statements A and B, we have determined that neither provides an accurate factorization of the polynomial 30x² + 40xy + 51y².

• Statement A incorrectly suggests factoring out 10, which does not account for the coefficient 51 in the original polynomial.
• Statement B proposes an impossible factorization form involving the product of a trinomial and xy, which would lead to terms with higher degrees than those in the original polynomial.

Given these findings, it is clear that the correct factorization method for 30x² + 40xy + 51y² requires a different approach. The polynomial does not have a simple factorization using integer coefficients. Attempts to factor it using common techniques such as GCF or trial and error will not yield integer factors.

Implications for Further Analysis

The fact that this polynomial does not factor easily has significant implications for solving equations or simplifying expressions that involve it. In such cases, alternative methods such as the quadratic formula or completing the square may be necessary, especially if we are looking for roots or solutions involving this polynomial.

• Quadratic Formula: For a quadratic equation of the form ax² + bx + c = 0, the quadratic formula provides a direct method for finding the roots. While our polynomial is in two variables, understanding the principles of the quadratic formula can be useful in related contexts.
• Completing the Square: This technique transforms a quadratic expression into a perfect square trinomial, which can simplify the process of finding roots or analyzing the expression's behavior. Though not directly applicable for factoring with integer coefficients in this case, the method is essential for broader algebraic manipulations.

In conclusion, the exercise of evaluating statements about polynomial factorization underscores the importance of understanding different factorization techniques and their limitations. The polynomial 30x² + 40xy + 51y² serves as a compelling example of a case where standard factorization methods do not lead to a simple factorization, highlighting the need for alternative approaches in more complex algebraic problems.

In this detailed analysis, we have shown that neither statement A nor statement B accurately describes the factorization of the polynomial 30x² + 40xy + 51y². This exploration reinforces the importance of careful examination and application of correct factorization techniques in algebra. The complexity of this polynomial serves as a valuable lesson in recognizing when standard methods may not suffice and when alternative approaches are necessary.