Factoring 2s^2 - 9s + 9 A Step-by-Step Guide
Factoring polynomials is a fundamental skill in algebra, essential for solving equations, simplifying expressions, and understanding the behavior of functions. In this article, we will delve into the process of factoring the quadratic polynomial 2s² - 9s + 9. We will explore various techniques and strategies to arrive at the correct factorization, providing a step-by-step guide to make the process clear and understandable. Whether you are a student learning algebra or someone looking to refresh your factoring skills, this article will provide a comprehensive understanding of how to factor this specific polynomial and similar quadratic expressions.
Understanding Quadratic Polynomials
Before we dive into the specifics of factoring 2s² - 9s + 9, let's establish a foundational understanding of quadratic polynomials. A quadratic polynomial is a polynomial of degree two, meaning the highest power of the variable is two. The general form of a quadratic polynomial is ax² + bx + c, where a, b, and c are constants, and x is the variable. In our case, the polynomial 2s² - 9s + 9 fits this form perfectly, with a = 2, b = -9, and c = 9. Recognizing this structure is the first step towards successfully factoring the polynomial.
Quadratic polynomials can be factored into two binomials, which are expressions of the form (px + q) and (rx + s), where p, q, r, and s are constants. The goal of factoring is to find these binomials such that their product is the original quadratic polynomial. In other words, we want to find (ps + q)(rx + s) = ax² + bx + c. This process involves identifying the correct constants that satisfy this equation. Factoring quadratic polynomials is crucial in various mathematical contexts, including solving quadratic equations, simplifying algebraic expressions, and graphing quadratic functions. Mastering this skill opens doors to more advanced topics in algebra and calculus.
The coefficients a, b, and c play critical roles in determining the factors of the quadratic polynomial. The leading coefficient, a, affects the coefficients of the variable terms in the binomial factors. The constant term, c, influences the constant terms in the binomial factors. The middle coefficient, b, represents the sum of the products of the constants in the binomial factors. By carefully analyzing these coefficients, we can systematically find the correct factors. Understanding the relationships between these coefficients and the factors is key to efficient and accurate factoring.
Methods for Factoring Quadratic Polynomials
There are several methods for factoring quadratic polynomials, each with its own strengths and applications. For 2s² - 9s + 9, we will focus on the AC method, which is particularly useful when the leading coefficient (the coefficient of the s² term) is not equal to 1. The AC method involves multiplying the leading coefficient (a) by the constant term (c), finding factors of this product that add up to the middle coefficient (b), and then rewriting the middle term using these factors. This method provides a structured approach to factoring, especially for more complex quadratic polynomials.
Another common method is the trial and error method, where we guess and check different combinations of binomial factors until we find the correct pair that multiplies to the original quadratic polynomial. While this method can be effective for simpler polynomials, it can become time-consuming and challenging for more complex expressions. The AC method offers a more systematic way to approach factoring, reducing the reliance on guesswork. However, understanding both methods can be beneficial, as different problems may be better suited to different approaches. Practicing both methods can enhance your overall factoring skills and problem-solving abilities.
A third method, particularly useful for perfect square trinomials, involves recognizing specific patterns in the quadratic polynomial. A perfect square trinomial is a quadratic polynomial that can be factored into the square of a binomial. For example, x² + 2xy + y² can be factored as (x + y)². While this method is not directly applicable to 2s² - 9s + 9, understanding the concept of perfect square trinomials can aid in recognizing and factoring other types of quadratic polynomials more efficiently. By mastering different factoring methods, you can develop a versatile skill set that allows you to tackle a wide range of algebraic problems.
Applying the AC Method to 2s² - 9s + 9
Now, let's apply the AC method to factor the polynomial 2s² - 9s + 9. The first step is to identify the coefficients a, b, and c. In this case, a = 2, b = -9, and c = 9. Next, we multiply a and c: 2 * 9 = 18. We then need to find two numbers that multiply to 18 and add up to b, which is -9. This is a crucial step, as the correct factors will lead us to the correct factorization of the quadratic polynomial.
To find the two numbers, we can list the factor pairs of 18: (1, 18), (2, 9), and (3, 6). Since we need the numbers to add up to -9, we need to consider the negative factors. The factor pairs become: (-1, -18), (-2, -9), and (-3, -6). The pair that adds up to -9 is (-3, -6). Now that we have found these two numbers, we can rewrite the middle term of the polynomial using these factors. This process is the core of the AC method and allows us to break down the quadratic polynomial into a form that can be factored by grouping.
We rewrite the polynomial 2s² - 9s + 9 as 2s² - 6s - 3s + 9. Notice that -6s - 3s is equivalent to -9s, so we have not changed the value of the polynomial. We have simply rewritten it in a way that allows us to factor by grouping. This technique is a key element of the AC method and is essential for factoring quadratic polynomials with a leading coefficient not equal to 1. The next step is to group the terms and factor out the greatest common factor (GCF) from each group.
Factoring by Grouping
After rewriting the polynomial as 2s² - 6s - 3s + 9, we can now factor by grouping. Grouping involves pairing the first two terms and the last two terms together: (2s² - 6s) + (-3s + 9). Next, we find the greatest common factor (GCF) for each group. For the first group, 2s² - 6s, the GCF is 2s. Factoring out 2s gives us 2s(s - 3). For the second group, -3s + 9, the GCF is -3. Factoring out -3 gives us -3(s - 3). It's important to factor out the negative sign in the second group to ensure that the binomial factors match, which is a critical step in factoring by grouping.
Now we have 2s(s - 3) - 3(s - 3). Notice that both terms have a common factor of (s - 3). We can factor out this common binomial factor to obtain (s - 3)(2s - 3). This is the factored form of the original quadratic polynomial 2s² - 9s + 9. Factoring by grouping is a powerful technique that allows us to handle more complex quadratic polynomials by breaking them down into smaller, more manageable parts. The key is to correctly identify the GCF in each group and to ensure that the binomial factors match after factoring out the GCF.
By successfully factoring by grouping, we have arrived at the factored form of the polynomial. This process demonstrates the effectiveness of the AC method in handling quadratic polynomials with a leading coefficient not equal to 1. The factored form, (s - 3)(2s - 3), provides valuable insights into the roots of the corresponding quadratic equation and the behavior of the quadratic function. The ability to factor polynomials is a fundamental skill that underpins many advanced concepts in algebra and calculus.
Verifying the Factorization
To ensure our factorization is correct, we can multiply the binomial factors (s - 3)(2s - 3) back together to see if we obtain the original polynomial 2s² - 9s + 9. This process is known as expanding the binomial factors and serves as a crucial verification step in factoring. By performing this check, we can confidently confirm that our factorization is accurate and that we have correctly applied the AC method and factoring by grouping.
Expanding the binomial factors involves using the distributive property (also known as the FOIL method) to multiply each term in the first binomial by each term in the second binomial. We have:
(s - 3)(2s - 3) = s(2s) + s(-3) - 3(2s) - 3(-3)
Simplifying each term, we get:
2s² - 3s - 6s + 9
Combining like terms, we have:
2s² - 9s + 9
This is indeed the original polynomial, which confirms that our factorization is correct. The verification step is essential because it helps us catch any potential errors in our factoring process. By taking the time to expand the binomial factors and compare the result to the original polynomial, we can be confident in our solution and avoid mistakes in further calculations or applications.
Verifying the factorization is not only a check for accuracy but also a way to deepen our understanding of the relationship between the factored form and the expanded form of a polynomial. It reinforces the concept that factoring and expanding are inverse operations, and mastering both skills is crucial for success in algebra. This process provides a concrete way to see how the binomial factors multiply together to create the original quadratic polynomial, enhancing our intuition and problem-solving abilities.
Final Answer and Conclusion
In conclusion, the factored form of the polynomial 2s² - 9s + 9 is (s - 3)(2s - 3). We arrived at this answer by applying the AC method, rewriting the middle term, factoring by grouping, and verifying our result by expanding the binomial factors. This process demonstrates the power and effectiveness of algebraic techniques in solving mathematical problems.
The ability to factor polynomials is a fundamental skill with wide-ranging applications in mathematics and other fields. It is essential for solving quadratic equations, simplifying algebraic expressions, graphing functions, and tackling more advanced topics in calculus and beyond. By mastering factoring techniques, you equip yourself with a powerful tool for problem-solving and analytical thinking.
We hope this comprehensive guide has provided you with a clear understanding of how to factor the polynomial 2s² - 9s + 9 and similar quadratic expressions. Remember to practice these techniques regularly to build your skills and confidence in factoring. With consistent effort, you can become proficient in factoring and unlock the doors to more advanced mathematical concepts.
