Factoring is a crucial math concept that helps us simplify expressions and solve equations. In this article, we'll delve into the step-by-step solution for factoring the expression 12x4 + 39x3 + 9x2.
The Importance of Factoring in Math
Factoring is an essential skill in mathematics, allowing us to break down complex expressions into simpler ones. By factoring, we can:
- Simplify expressions and make them easier to work with
- Solve equations and inequalities
- Identify common factors and patterns in expressions
- Enhance problem-solving skills and critical thinking
Why Factor 12x4 + 39x3 + 9x2?
The given expression, 12x4 + 39x3 + 9x2, is a polynomial expression with three terms. Factoring this expression will help us:
- Identify common factors and simplify the expression
- Solve equations and inequalities involving this expression
- Develop problem-solving skills and critical thinking
Step-by-Step Solution: Factoring 12x4 + 39x3 + 9x2
To factor the expression 12x4 + 39x3 + 9x2, we'll follow these steps:
Step 1: Factor out the Greatest Common Factor (GCF)
The first step in factoring is to identify the greatest common factor (GCF) of the terms. The GCF is the largest factor that divides all the terms evenly.
In this case, the GCF of 12x4, 39x3, and 9x2 is 3x2.
Step 2: Factor out the Common Binomial Factor
After factoring out the GCF, we're left with 4x2 + 13x + 3. Now, we need to factor out the common binomial factor, if any.
Upon inspection, we notice that 4x2 + 13x + 3 can be factored as (4x + 3)(x + 1).
Step 3: Write the Final Factored Form
Combining the GCF and the common binomial factor, we get:
12x4 + 39x3 + 9x2 = 3x2(4x + 3)(x + 1)
And that's the final factored form of the expression!
Key Takeaways
- Factoring is an essential skill in mathematics that helps simplify expressions and solve equations.
- The greatest common factor (GCF) is the largest factor that divides all the terms evenly.
- Factoring out the GCF and common binomial factor can help simplify complex expressions.
- The final factored form of 12x4 + 39x3 + 9x2 is 3x2(4x + 3)(x + 1).
Example Problems
Try factoring these expressions:
- 15x3 + 20x2 + 5x
- 24x4 + 30x3 + 6x2
Common Factoring Mistakes
When factoring, it's easy to make mistakes. Here are some common ones to watch out for:
- Forgetting to factor out the GCF
- Factoring out the wrong binomial factor
- Not simplifying the expression fully
Best Practices for Factoring
To improve your factoring skills, follow these best practices:
- Always factor out the GCF first
- Look for common binomial factors
- Simplify the expression fully
Real-World Applications of Factoring
Factoring has numerous real-world applications, including:
- Solving equations and inequalities in physics and engineering
- Simplifying expressions in computer science and coding
- Identifying patterns in data analysis and statistics
Conclusion
Factoring is a vital math concept that helps us simplify expressions and solve equations. By following the step-by-step solution for factoring 12x4 + 39x3 + 9x2, you've developed your problem-solving skills and critical thinking.
Remember to always factor out the GCF, look for common binomial factors, and simplify the expression fully. With practice and patience, you'll become a factoring pro in no time!
What are some common factoring mistakes you've encountered? Share your experiences and tips in the comments below!
What is the greatest common factor (GCF) of 12x4, 39x3, and 9x2?
+The GCF of 12x4, 39x3, and 9x2 is 3x2.
How do I factor out the common binomial factor?
+To factor out the common binomial factor, look for two terms that have a common factor. In this case, 4x2 + 13x + 3 can be factored as (4x + 3)(x + 1).
What are some real-world applications of factoring?
+Factoring has numerous real-world applications, including solving equations and inequalities in physics and engineering, simplifying expressions in computer science and coding, and identifying patterns in data analysis and statistics.