Factor The Polynomial $30x^2 - 23xy + 3y^2$.
Factoring polynomials is a fundamental skill in algebra, enabling us to simplify expressions, solve equations, and gain deeper insights into mathematical relationships. In this comprehensive guide, we will delve into the process of factoring the quadratic polynomial $30x^2 - 23xy + 3y^2$, providing a step-by-step approach and exploring the underlying concepts. This skill is crucial for students and anyone involved in mathematics, offering a robust method for problem-solving and algebraic manipulation.
Understanding Quadratic Polynomials
Before we dive into factoring the specific polynomial, it's essential to understand the structure of a quadratic polynomial. A quadratic polynomial is an expression of the form $ax^2 + bxy + cy^2$, where a, b, and c are constants, and x and y are variables. The goal of factoring is to express this polynomial as a product of two binomials. This process is the reverse of expansion, where we multiply binomials to obtain a quadratic polynomial. Factoring is a critical skill in algebra, simplifying expressions and revealing underlying structures.
In our case, the polynomial $30x^2 - 23xy + 3y^2$ fits this form, with a = 30, b = -23, and c = 3. The challenge is to find two binomials that, when multiplied, yield the original polynomial. Mastery of this technique enhances problem-solving capabilities in various mathematical contexts. The ability to recognize and factor quadratic polynomials is a cornerstone of algebraic proficiency. Understanding the components and structure of these polynomials is crucial for successful manipulation and simplification.
Step-by-Step Factoring Process
To factor the polynomial $30x^2 - 23xy + 3y^2$, we'll use a systematic approach. This involves finding two binomials of the form $(Ax + By)(Cx + Dy)$ such that when multiplied, they give us the original polynomial. The key is to find the correct coefficients A, B, C, and D.
1. Identify the Coefficients
First, we identify the coefficients of the quadratic polynomial: a = 30, b = -23, and c = 3. These coefficients will guide us in finding the appropriate factors. Accurate identification of these coefficients is the first critical step in the factoring process. This sets the foundation for subsequent steps and ensures that the factorization proceeds correctly. Attention to detail at this stage can prevent errors and streamline the solution.
2. Find Two Numbers
The next step is to find two numbers that multiply to ac (30 * 3 = 90) and add up to b (-23). This is a crucial step in the factoring process, often requiring some trial and error. We need to find two numbers whose product is 90 and whose sum is -23. This involves considering different factor pairs of 90 and checking their sums. The correct pair will enable us to rewrite the middle term and proceed with factoring by grouping. This is a critical step that transforms the trinomial into a form that can be factored more easily. By systematically testing factor pairs, we can efficiently find the appropriate numbers to continue the factorization process.
After some thought, we find that -18 and -5 satisfy these conditions because (-18) * (-5) = 90 and (-18) + (-5) = -23. The correct identification of these numbers is essential for the next steps in the factoring process. These numbers allow us to rewrite the middle term of the polynomial, which is a crucial step in factoring by grouping. Choosing the correct pair ensures the polynomial can be successfully factored into binomial expressions.
3. Rewrite the Middle Term
Now, we rewrite the middle term (-23xy) using the two numbers we found: -18xy and -5xy. The polynomial becomes: $30x^2 - 18xy - 5xy + 3y^2$. Rewriting the middle term in this way sets up the polynomial for factoring by grouping. This step is crucial because it allows us to break down the trinomial into a four-term expression, which can then be factored more easily. The strategic rewriting of the middle term is a key technique in factoring quadratic polynomials.
4. Factor by Grouping
Next, we group the terms in pairs and factor out the greatest common factor (GCF) from each pair:
- From the first two terms, $30x^2 - 18xy$, we can factor out 6x: $6x(5x - 3y)$.
- From the last two terms, $-5xy + 3y^2$, we can factor out -y: $-y(5x - 3y)$.
This gives us: $6x(5x - 3y) - y(5x - 3y)$. Factoring by grouping involves identifying common factors within pairs of terms, which simplifies the expression. This technique is particularly useful when dealing with four-term polynomials, as it allows us to consolidate the expression into a product of binomials. By systematically extracting the greatest common factor from each group, we prepare the polynomial for its final factored form. This method highlights the structure of the polynomial and facilitates its decomposition into simpler components. Factoring by grouping is a powerful technique in algebra, enabling us to handle complex expressions more effectively.
5. Factor out the Common Binomial
We notice that $(5x - 3y)$ is a common binomial factor in both terms. Factoring this out, we get: $(6x - y)(5x - 3y)$. The presence of a common binomial factor indicates that the polynomial is factorable, and extracting this common factor is the final step in the process. This step consolidates the expression into its fully factored form, revealing the underlying structure of the polynomial. Recognizing and factoring out the common binomial is a key skill in algebraic manipulation, leading to simplified expressions and solutions.
The Solution
Therefore, the factored form of the polynomial $30x^2 - 23xy + 3y^2$ is $(6x - y)(5x - 3y)$.
Verification
To verify our solution, we can multiply the binomials $(6x - y)$ and $(5x - 3y)$ using the distributive property (also known as the FOIL method):
This confirms that our factored form is correct. Verification ensures accuracy and provides confidence in the solution. By expanding the factored form, we can directly compare it to the original polynomial, confirming that they are equivalent. This step is crucial in mathematical problem-solving, as it serves as a check for errors and reinforces understanding of the factoring process.
Conclusion
Factoring the polynomial $30x^2 - 23xy + 3y^2$ involves a systematic approach, including identifying coefficients, finding the appropriate numbers, rewriting the middle term, factoring by grouping, and extracting the common binomial factor. The correct answer is **c. $(6x-y)(5x-3y)$. Mastering these techniques is crucial for success in algebra and beyond. Factoring is a fundamental skill in algebra, enabling the simplification of expressions and the solution of equations. A systematic approach, as demonstrated in this guide, ensures accuracy and efficiency in the factoring process. By understanding the underlying principles and practicing the steps involved, one can confidently tackle various factoring problems. This skill is not only valuable in mathematics but also in various fields that utilize algebraic concepts.
This comprehensive guide has walked you through the entire process, ensuring you have a solid understanding of how to factor quadratic polynomials. Through careful steps and verification, we've shown how to accurately factor $30x^2 - 23xy + 3y^2$.