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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Understanding the Problem
We are given a cubic expression in the form of $x^3+216$ and asked to find its factored form. Factoring a cubic expression involves expressing it as a product of three binomials. In this case, we need to find the correct factorization of the given expression.
The Correct Factorization
To factor the given expression, we can use the sum of cubes formula, which states that $a3+b3=(a+b)(a2-ab+b2)$. In this case, we have $x^3+216$, which can be written as $(x)3+(6)3$. Applying the sum of cubes formula, we get:
Evaluating the Options
Now that we have the correct factorization, let's evaluate the options given:
A. $(x-6)\left(x^2+6x+36\right)$
This option is incorrect because the correct factorization has a positive sign in the first binomial, not a negative sign.
B. $(x+6)\left(x^2-6x+36\right)$
This option is correct because it matches the factorization we obtained using the sum of cubes formula.
C. $(x+6)\left(x^2-12x+36\right)$
This option is incorrect because the quadratic term in the second binomial has a negative coefficient, whereas the correct factorization has a positive coefficient.
D. $(x-6)\left(x^2+6x+36\right)$
This option is incorrect because it has a negative sign in the first binomial, whereas the correct factorization has a positive sign.
Conclusion
In conclusion, the correct factorization of the given cubic expression $x^3+216$ is $(x+6)\left(x^2-6x+36\right)$. This can be obtained using the sum of cubes formula, which states that $a3+b3=(a+b)(a2-ab+b2)$. We evaluated the options given and found that only option B matches the correct factorization.
Tips and Tricks
When factoring a cubic expression, it's essential to look for a pattern that matches the sum of cubes formula. In this case, we had $(x)3+(6)3$, which made it easy to apply the formula. Additionally, we need to pay attention to the signs in the binomials and the quadratic term to ensure that we get the correct factorization.
Common Mistakes
One common mistake when factoring a cubic expression is to forget to use the sum of cubes formula. Another mistake is to get the signs in the binomials and the quadratic term wrong. To avoid these mistakes, it's crucial to carefully read the problem and apply the formula correctly.
Real-World Applications
Factoring a cubic expression has numerous real-world applications, such as:
- Engineering: Factoring cubic expressions is essential in engineering to solve problems involving forces, velocities, and accelerations.
- Physics: Factoring cubic expressions is used to solve problems involving energy, momentum, and motion.
- Computer Science: Factoring cubic expressions is used in computer science to solve problems involving algorithms and data structures.
Practice Problems
To practice factoring cubic expressions, try the following problems:
- Factor the expression $x^3+125$.
- Factor the expression $x^3-27$.
- Factor the expression $x^3+8$.
Conclusion
In conclusion, factoring a cubic expression involves expressing it as a product of three binomials. We used the sum of cubes formula to factor the given expression $x^3+216$ and obtained the correct factorization $(x+6)\left(x^2-6x+36\right)$. We evaluated the options given and found that only option B matches the correct factorization.
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Q: What is the sum of cubes formula?
A: The sum of cubes formula is a mathematical formula that states that $a3+b3=(a+b)(a2-ab+b2)$. This formula is used to factor a cubic expression of the form $a3+b3$.
Q: How do I apply the sum of cubes formula?
A: To apply the sum of cubes formula, you need to identify the values of $a$ and $b$ in the given expression. Then, you can plug these values into the formula to obtain the factored form.
Q: What are some common mistakes to avoid when factoring cubic expressions?
A: Some common mistakes to avoid when factoring cubic expressions include:
- Forgetting to use the sum of cubes formula
- Getting the signs in the binomials and the quadratic term wrong
- Not paying attention to the pattern of the expression
Q: How do I determine the correct factorization of a cubic expression?
A: To determine the correct factorization of a cubic expression, you need to carefully read the problem and apply the sum of cubes formula correctly. You should also pay attention to the signs in the binomials and the quadratic term.
Q: What are some real-world applications of factoring cubic expressions?
A: Factoring cubic expressions has numerous real-world applications, including:
- Engineering: Factoring cubic expressions is essential in engineering to solve problems involving forces, velocities, and accelerations.
- Physics: Factoring cubic expressions is used to solve problems involving energy, momentum, and motion.
- Computer Science: Factoring cubic expressions is used in computer science to solve problems involving algorithms and data structures.
Q: How can I practice factoring cubic expressions?
A: You can practice factoring cubic expressions by trying the following problems:
- Factor the expression $x^3+125$.
- Factor the expression $x^3-27$.
- Factor the expression $x^3+8$.
Q: What are some tips for factoring cubic expressions?
A: Some tips for factoring cubic expressions include:
- Paying attention to the pattern of the expression
- Using the sum of cubes formula correctly
- Double-checking your work to ensure that you get the correct factorization
Q: How do I know if I have factored a cubic expression correctly?
A: To know if you have factored a cubic expression correctly, you need to check your work carefully. You should also use a calculator or a computer program to verify your answer.
Q: What are some common errors to watch out for when factoring cubic expressions?
A: Some common errors to watch out for when factoring cubic expressions include:
- Forgetting to use the sum of cubes formula
- Getting the signs in the binomials and the quadratic term wrong
- Not paying attention to the pattern of the expression
Q: How can I use factoring to solve problems in real-world applications?
A: You can use factoring to solve problems in real-world applications by applying the sum of cubes formula and paying attention to the signs in the binomials and the quadratic term. You can also use factoring to solve problems involving forces, velocities, and accelerations in engineering, energy, momentum, and motion in physics, and algorithms and data structures in computer science.
Q: What are some advanced topics in factoring cubic expressions?
A: Some advanced topics in factoring cubic expressions include:
- Factoring expressions with complex numbers
- Factoring expressions with rational exponents
- Factoring expressions with radical expressions
Q: How can I learn more about factoring cubic expressions?
A: You can learn more about factoring cubic expressions by:
- Reading textbooks and online resources
- Watching video tutorials and online lectures
- Practicing problems and exercises
- Joining online communities and forums
Q: What are some resources for learning factoring cubic expressions?
A: Some resources for learning factoring cubic expressions include:
- Textbooks and online resources
- Video tutorials and online lectures
- Practice problems and exercises
- Online communities and forums
Q: How can I apply factoring to solve problems in mathematics and science?
A: You can apply factoring to solve problems in mathematics and science by using the sum of cubes formula and paying attention to the signs in the binomials and the quadratic term. You can also use factoring to solve problems involving forces, velocities, and accelerations in engineering, energy, momentum, and motion in physics, and algorithms and data structures in computer science.
Q: What are some examples of factoring cubic expressions in real-world applications?
A: Some examples of factoring cubic expressions in real-world applications include:
- Factoring expressions involving forces and velocities in engineering
- Factoring expressions involving energy and momentum in physics
- Factoring expressions involving algorithms and data structures in computer science
Q: How can I use factoring to solve problems in algebra and geometry?
A: You can use factoring to solve problems in algebra and geometry by applying the sum of cubes formula and paying attention to the signs in the binomials and the quadratic term. You can also use factoring to solve problems involving equations and inequalities in algebra and shapes and figures in geometry.
Q: What are some tips for factoring expressions with complex numbers?
A: Some tips for factoring expressions with complex numbers include:
- Using the sum of cubes formula correctly
- Paying attention to the signs in the binomials and the quadratic term
- Double-checking your work to ensure that you get the correct factorization
Q: How can I apply factoring to solve problems in calculus and differential equations?
A: You can apply factoring to solve problems in calculus and differential equations by using the sum of cubes formula and paying attention to the signs in the binomials and the quadratic term. You can also use factoring to solve problems involving limits and derivatives in calculus and differential equations.
Q: What are some examples of factoring cubic expressions in calculus and differential equations?
A: Some examples of factoring cubic expressions in calculus and differential equations include:
- Factoring expressions involving limits and derivatives in calculus
- Factoring expressions involving differential equations in differential equations
Q: How can I use factoring to solve problems in statistics and probability?
A: You can use factoring to solve problems in statistics and probability by applying the sum of cubes formula and paying attention to the signs in the binomials and the quadratic term. You can also use factoring to solve problems involving data analysis and probability distributions in statistics and probability.
Q: What are some tips for factoring expressions with rational exponents?
A: Some tips for factoring expressions with rational exponents include:
- Using the sum of cubes formula correctly
- Paying attention to the signs in the binomials and the quadratic term
- Double-checking your work to ensure that you get the correct factorization
Q: How can I apply factoring to solve problems in finance and economics?
A: You can apply factoring to solve problems in finance and economics by using the sum of cubes formula and paying attention to the signs in the binomials and the quadratic term. You can also use factoring to solve problems involving financial models and economic systems in finance and economics.
Q: What are some examples of factoring cubic expressions in finance and economics?
A: Some examples of factoring cubic expressions in finance and economics include:
- Factoring expressions involving financial models and economic systems in finance and economics
Q: How can I use factoring to solve problems in environmental science and ecology?
A: You can use factoring to solve problems in environmental science and ecology by applying the sum of cubes formula and paying attention to the signs in the binomials and the quadratic term. You can also use factoring to solve problems involving environmental systems and ecosystems in environmental science and ecology.
Q: What are some tips for factoring expressions with radical expressions?
A: Some tips for factoring expressions with radical expressions include:
- Using the sum of cubes formula correctly
- Paying attention to the signs in the binomials and the quadratic term
- Double-checking your work to ensure that you get the correct factorization
Q: How can I apply factoring to solve problems in social sciences and humanities?
A: You can apply factoring to solve problems in social sciences and humanities by using the sum of cubes formula and paying attention to the signs in the binomials and the quadratic term. You can also use factoring to solve problems involving social systems and cultural phenomena in social sciences and humanities.
Q: What are some examples of factoring cubic expressions in social sciences and humanities?
A: Some examples of factoring cubic expressions in social sciences and humanities include:
- Factoring expressions involving social systems and cultural phenomena in social sciences and humanities
Q: How can I use factoring to solve problems in computer science and information technology?
A: You can use factoring to solve problems in computer science and information technology by applying the sum of cubes formula and paying attention to the signs in the bin