Factoring Completely A Step-by-Step Guide To 6x² - 3x - 30
Factoring polynomials is a fundamental skill in algebra, serving as a cornerstone for solving equations, simplifying expressions, and understanding the behavior of functions. In this comprehensive guide, we will delve into the process of factoring the quadratic expression 6x² - 3x - 30 completely. We will break down each step, providing clear explanations and strategies to ensure you grasp the underlying concepts. By the end of this guide, you will be equipped with the knowledge and confidence to tackle similar factoring problems with ease.
Understanding Factoring
Before we dive into the specifics of factoring 6x² - 3x - 30, let's establish a solid understanding of what factoring entails. Factoring, in essence, is the reverse process of multiplication. When we multiply two or more expressions, we expand them to obtain a single, more complex expression. Factoring, on the other hand, involves breaking down a complex expression into its constituent factors, which are the expressions that multiply together to yield the original expression. Imagine it like disassembling a machine into its individual components – factoring is the process of taking a complex polynomial and identifying its simpler building blocks.
In the realm of polynomials, factoring is particularly useful for solving equations. The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property forms the basis for solving factored polynomial equations. By factoring a polynomial and setting each factor equal to zero, we can find the solutions or roots of the equation.
Step-by-Step Factoring of 6x² - 3x - 30
Now, let's embark on the journey of factoring the quadratic expression 6x² - 3x - 30 completely. We will follow a systematic approach, ensuring that we identify and extract all common factors, as well as factor the resulting quadratic expression if possible.
Step 1: Identify the Greatest Common Factor (GCF)
The first step in factoring any polynomial is to identify the Greatest Common Factor (GCF) of all the terms. The GCF is the largest factor that divides evenly into each term of the polynomial. In the expression 6x² - 3x - 30, we need to find the largest factor that divides evenly into 6, -3, and -30. By inspection, we can see that the GCF is 3. Factoring out the GCF, we get:
6x² - 3x - 30 = 3(2x² - x - 10)
This step is crucial because it simplifies the expression and makes it easier to factor further. Always look for the GCF as the first step in factoring any polynomial.
Step 2: Factor the Quadratic Expression
After factoring out the GCF, we are left with the quadratic expression 2x² - x - 10. To factor this quadratic, we can use several methods, such as the trial and error method, the AC method, or the quadratic formula. For this example, we will use the AC method.
The AC Method
The AC method involves the following steps:
-
Identify the coefficients a, b, and c in the quadratic expression ax² + bx + c. In our case, a = 2, b = -1, and c = -10.
-
Multiply a and c: 2 * -10 = -20.
-
Find two factors of ac (-20) that add up to b (-1). The factors -5 and 4 satisfy this condition since -5 * 4 = -20 and -5 + 4 = -1.
-
Rewrite the middle term (-x) using the two factors we found (-5x and 4x): 2x² - 5x + 4x - 10.
-
Factor by grouping: Group the first two terms and the last two terms and factor out the GCF from each group:
(2x² - 5x) + (4x - 10) = x(2x - 5) + 2(2x - 5) -
Factor out the common binomial factor (2x - 5):
x(2x - 5) + 2(2x - 5) = (2x - 5)(x + 2)
Thus, the factored form of 2x² - x - 10 is (2x - 5)(x + 2).
Step 3: Combine the Factors
Now that we have factored both the GCF and the quadratic expression, we need to combine them to obtain the complete factored form of the original expression:
6x² - 3x - 30 = 3(2x² - x - 10) = 3(2x - 5)(x + 2)
Therefore, the completely factored form of 6x² - 3x - 30 is 3(2x - 5)(x + 2).
Alternative Factoring Methods
While we used the AC method to factor the quadratic expression, other methods can also be employed. Let's briefly explore two alternative methods:
Trial and Error Method
The trial and error method, also known as the guess and check method, involves systematically trying different combinations of factors until the correct combination is found. This method can be effective for simpler quadratic expressions, but it can become more challenging for expressions with larger coefficients or more complex factoring patterns. To use the trial and error method, we would list out all possible factors of the leading coefficient (2) and the constant term (-10) and try different combinations until we find a combination that produces the correct middle term (-x).
Quadratic Formula
The quadratic formula is a general formula that can be used to find the roots of any quadratic equation of the form ax² + bx + c = 0. The formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
While the quadratic formula doesn't directly factor the expression, it can help us find the roots, which can then be used to determine the factors. If the roots are r₁ and r₂, then the factors of the quadratic expression are (x - r₁) and (x - r₂). However, the quadratic formula is typically used when the quadratic expression is difficult to factor using other methods.
Placing the Answer on the Grid
The final step in the original prompt involves placing the factored answer, 3(2x - 5)(x + 2), on a grid. The specific format of the grid and the instructions for placing the answer may vary depending on the context of the problem. However, the general idea is to represent the factored expression in a visual or structured format. This could involve writing each factor in a separate box, or using a grid to represent the coefficients and constants of the factors.
For example, if the grid consists of boxes for each factor, we could place the factors as follows:
- Box 1: 3
- Box 2: (2x - 5)
- Box 3: (x + 2)
Alternatively, if the grid is designed to represent the coefficients and constants, we would need to expand the factored expression to its standard form (6x² - 3x - 30) and then place the coefficients and constant in the corresponding grid cells.
Common Factoring Mistakes to Avoid
Factoring can be tricky, and it's easy to make mistakes if you're not careful. Here are some common factoring mistakes to avoid:
- Forgetting to factor out the GCF: Always look for the GCF as the first step in factoring. Failing to do so can make the remaining factoring process more difficult.
- Incorrectly identifying factors: Double-check your factors to make sure they multiply to give the correct product and add up to the correct sum.
- Sign errors: Pay close attention to the signs of the terms when factoring. A simple sign error can lead to an incorrect factored form.
- Stopping too early: Make sure you have factored the expression completely. This means that none of the factors can be factored further.
- Not checking your answer: After factoring, you can always check your answer by multiplying the factors back together. If the result is the original expression, then your factoring is correct.
Conclusion
Factoring polynomials is a crucial skill in algebra, with applications in various mathematical contexts. In this comprehensive guide, we have explored the process of factoring the quadratic expression 6x² - 3x - 30 completely. We have covered the importance of identifying the GCF, using the AC method to factor quadratic expressions, and combining the factors to obtain the final factored form. Additionally, we have discussed alternative factoring methods, common factoring mistakes to avoid, and the importance of placing the answer correctly on a grid.
By mastering the techniques and strategies outlined in this guide, you will be well-equipped to tackle a wide range of factoring problems with confidence and accuracy. Remember to practice regularly and apply these concepts to different types of polynomials to solidify your understanding. Happy factoring!
