Select The Correct Answer.What Is The Factored Form Of $x^3+216$?A. $(x-6)\left(x^2+6x+36\right)$B. $ ( X + 6 ) ( X 2 − 6 X + 36 ) (x+6)\left(x^2-6x+36\right) ( X + 6 ) ( X 2 − 6 X + 36 ) [/tex]C. $(x+6)\left(x^2-12x+36\right)$D.
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Introduction
Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. In this article, we will focus on factoring the cubic expression $x^3+216$ and explore the different methods and techniques used to achieve this.
Understanding the Cubic Expression
The given cubic expression is $x^3+216$. To factor this expression, we need to identify the greatest common factor (GCF) of the two terms. In this case, the GCF is 1, as there is no common factor between $x^3$ and 216.
Factoring by Grouping
One method of factoring a cubic expression is by grouping. This involves grouping the terms in pairs and factoring out the common factor from each pair.
Step 1: Group the Terms
Group the terms $x^3$ and 216 into two pairs: $x^3$ and $216$.
Step 2: Factor Out the Common Factor
Factor out the common factor from each pair. In this case, we can factor out $x^3$ from the first pair and 216 from the second pair.
Step 3: Factor the Difference of Cubes
Now that we have factored out the common factor, we can use the difference of cubes formula to factor the expression.
Factoring by Recognizing the Perfect Cube
Another method of factoring a cubic expression is by recognizing the perfect cube. A perfect cube is a number that can be expressed as the cube of an integer.
Step 1: Identify the Perfect Cube
In this case, we can recognize that 216 is a perfect cube, as it can be expressed as $6^3$.
Step 2: Factor the Perfect Cube
Now that we have identified the perfect cube, we can factor the expression using the formula for the difference of cubes.
Conclusion
In conclusion, we have factored the cubic expression $x^3+216$ using two different methods: factoring by grouping and factoring by recognizing the perfect cube. Both methods have led us to the same result: $(x+6)(x^2-6x+36)$.
Answer
The correct answer is:
- B. $(x+6)\left(x^2-6x+36\right)$
This is the factored form of the given cubic expression $x^3+216$.
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Introduction
In our previous article, we explored the different methods and techniques used to factor the cubic expression $x^3+216$. In this article, we will provide a Q&A guide to help you better understand the concept of factoring and how to apply it to different types of expressions.
Q&A: Factoring the Cubic Expression
Q: What is the greatest common factor (GCF) of the two terms in the cubic expression $x^3+216$?
A: The GCF of the two terms is 1, as there is no common factor between $x^3$ and 216.
Q: How do you factor a cubic expression by grouping?
A: To factor a cubic expression by grouping, you need to group the terms in pairs and factor out the common factor from each pair.
Q: What is the difference of cubes formula?
A: The difference of cubes formula is $(a-b)(a2+ab+b2)$.
Q: How do you factor a perfect cube?
A: To factor a perfect cube, you need to recognize the perfect cube and use the formula for the difference of cubes.
Q: What is the factored form of the cubic expression $x^3+216$?
A: The factored form of the cubic expression $x^3+216$ is $(x+6)(x^2-6x+36)$.
Q: What are some common mistakes to avoid when factoring a cubic expression?
A: Some common mistakes to avoid when factoring a cubic expression include:
- Not identifying the greatest common factor (GCF) of the two terms.
- Not grouping the terms correctly.
- Not factoring out the common factor from each pair.
- Not recognizing the perfect cube.
Tips and Tricks
Tip 1: Identify the Greatest Common Factor (GCF)
When factoring a cubic expression, it's essential to identify the greatest common factor (GCF) of the two terms. This will help you determine the correct factored form of the expression.
Tip 2: Group the Terms Correctly
When grouping the terms, make sure to group them in pairs and factor out the common factor from each pair.
Tip 3: Recognize the Perfect Cube
When factoring a perfect cube, make sure to recognize the perfect cube and use the formula for the difference of cubes.
Conclusion
In conclusion, factoring a cubic expression can be a challenging task, but with the right techniques and strategies, you can achieve the correct factored form. Remember to identify the greatest common factor (GCF), group the terms correctly, and recognize the perfect cube.
Common Mistakes to Avoid
When factoring a cubic expression, there are several common mistakes to avoid. These include:
- Not identifying the greatest common factor (GCF) of the two terms.
- Not grouping the terms correctly.
- Not factoring out the common factor from each pair.
- Not recognizing the perfect cube.
Final Answer
The final answer is:
- B. $(x+6)\left(x^2-6x+36\right)$
This is the factored form of the given cubic expression $x^3+216$.