Factoring 3x^2 - 11x + 8 A Step By Step Guide
Factoring quadratic expressions is a fundamental skill in algebra. In this comprehensive guide, we will walk you through the process of completely factoring the quadratic expression 3x^2 - 11x + 8. This skill is crucial for solving quadratic equations, simplifying algebraic fractions, and tackling various mathematical problems. Our goal is to provide a clear, step-by-step approach that will empower you to confidently factor similar expressions in the future. Understanding the nuances of factoring not only enhances your algebraic proficiency but also builds a solid foundation for more advanced mathematical concepts. Let's dive into the process and explore the techniques required to break down this quadratic expression into its simplest factors.
Understanding Quadratic Expressions
Before we delve into the factoring process, it's important to understand the general form of a quadratic expression and the terminology associated with it. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable is two. The standard form of a quadratic expression is ax^2 + bx + c, where 'a', 'b', and 'c' are constants, and 'x' is the variable. In our case, the expression 3x^2 - 11x + 8 fits this form, with a = 3, b = -11, and c = 8. Recognizing this standard form is the first step in understanding how to approach factoring. Each term plays a specific role: 'ax^2' is the quadratic term, 'bx' is the linear term, and 'c' is the constant term. The coefficients 'a', 'b', and 'c' dictate the shape and position of the parabola when the quadratic expression is graphed, and they also influence the way we factor the expression. Understanding the relationship between these coefficients is key to mastering factoring techniques.
Identifying the Factoring Method
When faced with a quadratic expression, the first step is to determine the most appropriate method for factoring. There are several techniques available, including factoring out the greatest common factor (GCF), using the difference of squares pattern, perfect square trinomials, and the ac method (also known as factoring by grouping). For the expression 3x^2 - 11x + 8, we observe that there is no GCF among the coefficients 3, -11, and 8, and the expression doesn't fit the difference of squares or perfect square trinomial patterns. Therefore, the ac method is the most suitable approach. This method is particularly useful when the coefficient of the x^2 term (a) is not equal to 1, as it provides a systematic way to find the factors. The ac method involves multiplying the coefficients 'a' and 'c', finding two numbers that multiply to this product and add up to 'b', and then using these numbers to rewrite the middle term and factor by grouping. This method ensures that we account for all the terms and their coefficients, leading to the correct factorization of the quadratic expression. Recognizing the structure of the expression and selecting the appropriate method is a critical step in the factoring process.
Applying the AC Method
The ac method is a powerful technique for factoring quadratic expressions, especially when the leading coefficient (a) is not 1. In our expression, 3x^2 - 11x + 8, we have a = 3, b = -11, and c = 8. The first step is to multiply a and c: 3 * 8 = 24. Next, we need to find two numbers that multiply to 24 and add up to b (-11). By considering the factors of 24, we identify -3 and -8 as the two numbers that satisfy these conditions because (-3) * (-8) = 24 and (-3) + (-8) = -11. These numbers will help us rewrite the middle term of the quadratic expression. We replace -11x with -3x - 8x, so the expression becomes 3x^2 - 3x - 8x + 8. Now, we have four terms, which allows us to factor by grouping. We group the first two terms and the last two terms: (3x^2 - 3x) + (-8x + 8). From the first group, we can factor out 3x, and from the second group, we can factor out -8. This gives us 3x(x - 1) - 8(x - 1). Notice that both terms now have a common factor of (x - 1), which we can factor out. This results in (x - 1)(3x - 8). Therefore, the factored form of 3x^2 - 11x + 8 is (x - 1)(3x - 8). This methodical approach ensures that we break down the quadratic expression correctly, leading to its simplest factors.
Verifying the Factors
After factoring a quadratic expression, it's crucial to verify the factors to ensure accuracy. A simple method to verify the factors is to multiply them back together and check if the result matches the original expression. In our case, we factored 3x^2 - 11x + 8 into (x - 1)(3x - 8). To verify, we multiply these two factors using the distributive property (also known as the FOIL method): (x - 1)(3x - 8) = x(3x) + x(-8) - 1(3x) - 1(-8). This simplifies to 3x^2 - 8x - 3x + 8. Combining the like terms, we get 3x^2 - 11x + 8, which is the original expression. This verification confirms that our factoring is correct. By multiplying the factors back together, we ensure that no errors were made during the factoring process. This step is an essential part of the problem-solving process, as it provides a reliable way to validate the solution and build confidence in your factoring skills. It also reinforces the understanding of how factors and the original expression are related.
The Complete Factorization
Through our step-by-step process, we have successfully factored the quadratic expression 3x^2 - 11x + 8. We identified the expression as a quadratic in the standard form ax^2 + bx + c, recognized that the ac method was the most suitable factoring technique, applied the ac method to find the factors, and verified our solution by multiplying the factors back together. The complete factorization of 3x^2 - 11x + 8 is (x - 1)(3x - 8). This means that the original quadratic expression can be written as the product of two binomial factors. Factoring quadratic expressions is a fundamental skill in algebra with wide-ranging applications. It is essential for solving quadratic equations, simplifying rational expressions, and understanding the behavior of quadratic functions. By mastering factoring techniques, you enhance your ability to manipulate algebraic expressions and solve complex mathematical problems. This example demonstrates a methodical approach to factoring, which can be applied to a variety of quadratic expressions. The key is to understand the underlying principles, choose the appropriate method, and meticulously carry out the steps to arrive at the correct factors.
Conclusion
In conclusion, the complete factorization of the quadratic expression 3x^2 - 11x + 8 is (x - 1)(3x - 8). We have demonstrated a systematic approach using the ac method, which is particularly effective for quadratics where the leading coefficient is not 1. This process involves multiplying the leading coefficient (a) and the constant term (c), finding two numbers that multiply to this product and add up to the coefficient of the linear term (b), rewriting the middle term, and then factoring by grouping. Verification by multiplying the factors back together is a critical step to ensure accuracy. Mastering these factoring techniques is essential for success in algebra and beyond. Factoring is not just a mathematical exercise; it's a powerful tool for solving equations, simplifying expressions, and gaining a deeper understanding of mathematical relationships. The ability to factor quadratic expressions confidently opens doors to more advanced mathematical concepts and applications. By practicing and applying these techniques, you can strengthen your algebraic skills and tackle a wide range of problems effectively. Remember, the key to mastering factoring lies in understanding the principles, choosing the right method, and meticulously executing each step. This example serves as a comprehensive guide to factoring quadratic expressions, empowering you to approach similar problems with confidence and accuracy.
