Factoring 8b² - 11b A Step-by-Step Guide
Factoring quadratic expressions is a fundamental skill in algebra, and mastering it opens doors to solving equations, simplifying expressions, and understanding mathematical relationships. In this comprehensive guide, we will delve into the process of factoring the quadratic expression 8b² - 11b. This expression is a classic example of factoring out the greatest common factor (GCF), a technique that forms the basis for more advanced factoring methods. We will break down the steps involved, explain the underlying principles, and provide examples to solidify your understanding. By the end of this guide, you will be well-equipped to factor similar expressions with confidence and precision.
Understanding Factoring
Before we dive into the specifics of factoring 8b² - 11b, it's crucial to understand the concept of factoring itself. At its core, factoring is the reverse process of multiplication. When we multiply two or more expressions, we expand them to obtain a single expression. Factoring, on the other hand, involves breaking down a single expression into its constituent factors – the expressions that, when multiplied together, produce the original expression. For instance, consider the number 12. We can factor it as 2 x 6 or 3 x 4, because both pairs of factors multiply to give 12. Similarly, in algebra, we factor expressions involving variables and constants.
The purpose of factoring is multifaceted. It allows us to simplify complex expressions, solve equations, and analyze the behavior of functions. In the context of solving quadratic equations, factoring is a powerful tool for finding the roots or solutions of the equation. By expressing the quadratic equation in its factored form, we can easily identify the values of the variable that make the equation equal to zero. Factoring also plays a vital role in simplifying rational expressions, which are fractions where the numerator and denominator are polynomials. By factoring both the numerator and denominator, we can cancel out common factors and reduce the expression to its simplest form. Furthermore, factoring is essential for understanding the graphs of polynomial functions. The factors of a polynomial equation correspond to the x-intercepts of its graph, providing valuable information about the function's behavior.
Identifying the Greatest Common Factor (GCF)
The first step in factoring any expression is to identify the greatest common factor (GCF) of its terms. The GCF is the largest factor that divides evenly into all the terms of the expression. In the case of 8b² - 11b, we need to find the GCF of the terms 8b² and -11b. To do this, we can break down each term into its prime factors.
The term 8b² can be expressed as 2 x 2 x 2 x b x b, while the term -11b can be expressed as -1 x 11 x b. By comparing the prime factors of both terms, we can identify the common factors. Both terms share a factor of 'b'. The numerical coefficients, 8 and -11, do not share any common factors other than 1. Therefore, the GCF of 8b² and -11b is 'b'. Identifying the GCF is a critical step because it allows us to factor out the largest possible expression, simplifying the remaining expression within the parentheses. This not only makes the factoring process easier but also ensures that the expression is factored completely.
Factoring Out the GCF
Now that we have identified the GCF of 8b² - 11b as 'b', we can proceed to factor it out. Factoring out the GCF involves dividing each term of the expression by the GCF and writing the GCF outside a set of parentheses, followed by the resulting expression inside the parentheses. In our case, we divide both 8b² and -11b by 'b'. Dividing 8b² by 'b' gives us 8b, and dividing -11b by 'b' gives us -11. Therefore, when we factor out 'b' from 8b² - 11b, we get b(8b - 11). This expression represents the factored form of the original expression. The term 'b' outside the parentheses is the GCF, and the expression (8b - 11) inside the parentheses is the result of dividing the original expression by the GCF.
To verify that we have factored correctly, we can distribute the GCF back into the parentheses. If we multiply 'b' by (8b - 11), we should obtain the original expression, 8b² - 11b. Performing the multiplication, we get b * 8b - b * 11, which simplifies to 8b² - 11b. This confirms that our factoring is correct. Factoring out the GCF is a crucial step in simplifying expressions and solving equations. It allows us to break down complex expressions into simpler factors, making them easier to work with. In the case of 8b² - 11b, factoring out the GCF 'b' results in the factored form b(8b - 11), which is the final factored form of the expression.
Checking Your Answer
As we briefly discussed earlier, checking your answer is an essential step in the factoring process. It ensures that you have factored the expression correctly and haven't made any mistakes along the way. The most common method for checking your factoring is to distribute the GCF back into the parentheses. If the result of the distribution matches the original expression, then your factoring is correct. In our case, we factored 8b² - 11b as b(8b - 11). To check our answer, we distribute 'b' back into the parentheses: b * 8b - b * 11 = 8b² - 11b. Since this matches our original expression, we can be confident that our factoring is correct.
Another way to check your factoring is to substitute a numerical value for the variable and evaluate both the original expression and the factored expression. If the two expressions yield the same result, then your factoring is likely correct. For example, let's substitute b = 2 into the original expression 8b² - 11b and the factored expression b(8b - 11). For the original expression, we have 8(2)² - 11(2) = 8(4) - 22 = 32 - 22 = 10. For the factored expression, we have 2(8(2) - 11) = 2(16 - 11) = 2(5) = 10. Since both expressions yield the same result (10), this further confirms that our factoring is correct. Checking your answer is a crucial habit to develop when factoring expressions. It helps you identify and correct any errors, ensuring that you have a solid understanding of the factoring process.
Examples and Practice Problems
To further solidify your understanding of factoring out the GCF, let's work through a few more examples. Consider the expression 6x² + 9x. The GCF of 6x² and 9x is 3x. Factoring out 3x, we get 3x(2x + 3). To check our answer, we distribute 3x back into the parentheses: 3x * 2x + 3x * 3 = 6x² + 9x, which matches the original expression.
Another example is the expression 12y³ - 18y². The GCF of 12y³ and -18y² is 6y². Factoring out 6y², we get 6y²(2y - 3). Checking our answer, we distribute 6y² back into the parentheses: 6y² * 2y - 6y² * 3 = 12y³ - 18y², which matches the original expression. Now, let's try a practice problem. Factor the expression 5a³ + 10a². What is the GCF of 5a³ and 10a²? It's 5a². Factoring out 5a², we get 5a²(a + 2). Check your answer by distributing 5a² back into the parentheses. You should get 5a³ + 10a², which confirms that your factoring is correct. Practice is key to mastering factoring. The more examples you work through, the more comfortable you will become with the process.
Conclusion
In this comprehensive guide, we have explored the process of factoring the expression 8b² - 11b. We began by understanding the concept of factoring and its importance in algebra. We then identified the greatest common factor (GCF) of the terms in the expression, which was 'b'. We factored out the GCF, resulting in the factored form b(8b - 11). We emphasized the importance of checking your answer by distributing the GCF back into the parentheses and verifying that it matches the original expression. We also worked through several examples and practice problems to solidify your understanding of factoring out the GCF.
Factoring is a fundamental skill in algebra, and mastering it will greatly enhance your ability to solve equations, simplify expressions, and analyze mathematical relationships. By understanding the principles of factoring and practicing regularly, you can develop confidence and proficiency in this essential skill. Remember, factoring is the reverse of multiplication, and the goal is to break down an expression into its constituent factors. The GCF is the largest factor that divides evenly into all the terms of the expression, and factoring it out simplifies the remaining expression. Always check your answer to ensure that you have factored correctly. With consistent practice, you will become adept at factoring expressions and applying this skill to solve a wide range of mathematical problems.
