Simplify (write ALL Your Steps):${ \begin{array}{l} (x+3)\left(x^2-3x+9\right) \ = X(x^2-3x+9) + 3(x^2-3x+9) \ = X^3 - 3x^2 + 9x + 3x^2 - 9x + 27 \ = X^3 + 27 \end{array} }$Is The Answer A Sum/difference Of Cubes? YES/NO

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Introduction

Algebraic expressions can be complex and daunting, but with a clear understanding of the steps involved in simplifying them, you can master this essential math skill. In this article, we will walk you through the process of simplifying a given algebraic expression, using the example of (x+3)(x2−3x+9)(x+3)\left(x^2-3x+9\right).

Step 1: Distribute the Terms

The first step in simplifying the given expression is to distribute the terms. This involves multiplying each term in the first expression by each term in the second expression.

(x+3)\left(x^2-3x+9\right) = x(x^2-3x+9) + 3(x^2-3x+9)

Step 2: Multiply the Terms

Next, we multiply the terms in each expression. This involves multiplying each term in the first expression by each term in the second expression.

x(x^2-3x+9) = x^3 - 3x^2 + 9x
3(x^2-3x+9) = 3x^2 - 9x + 27

Step 3: Combine Like Terms

Now, we combine like terms. This involves adding or subtracting terms that have the same variable and exponent.

x^3 - 3x^2 + 9x + 3x^2 - 9x + 27 = x^3 + 27

Step 4: Check for Sum/Difference of Cubes

The final expression is x3+27x^3 + 27. To determine if this is a sum or difference of cubes, we need to check if it can be written in the form a3+b3a^3 + b^3 or a3−b3a^3 - b^3.

x^3 + 27 = (x)^3 + (3)^3

Conclusion

The given expression (x+3)(x2−3x+9)(x+3)\left(x^2-3x+9\right) simplifies to x3+27x^3 + 27. This is a sum of cubes, as it can be written in the form a3+b3a^3 + b^3.

Discussion

Simplifying algebraic expressions is an essential skill in mathematics. By following the steps outlined in this article, you can master this skill and apply it to a wide range of problems. Remember to always distribute the terms, multiply the terms, combine like terms, and check for sum or difference of cubes.

Tips and Tricks

  • When simplifying algebraic expressions, always start by distributing the terms.
  • Use the FOIL method to multiply the terms in each expression.
  • Combine like terms by adding or subtracting terms that have the same variable and exponent.
  • Check for sum or difference of cubes by looking for terms that can be written in the form a3+b3a^3 + b^3 or a3−b3a^3 - b^3.

Examples

  • Simplify the expression (x−2)(x2+4x+4)(x-2)\left(x^2+4x+4\right).
  • Simplify the expression (x+1)(x2−2x+1)(x+1)\left(x^2-2x+1\right).
  • Simplify the expression (x−3)(x2+6x+9)(x-3)\left(x^2+6x+9\right).

Solutions

  • (x−2)(x2+4x+4)=x3+4x2+4x−2x2−8x−8=x3+2x2−4x−8(x-2)\left(x^2+4x+4\right) = x^3 + 4x^2 + 4x - 2x^2 - 8x - 8 = x^3 + 2x^2 - 4x - 8
  • (x+1)(x2−2x+1)=x3−2x2+x+x2−2x+1=x3−x2−x+1(x+1)\left(x^2-2x+1\right) = x^3 - 2x^2 + x + x^2 - 2x + 1 = x^3 - x^2 - x + 1
  • (x−3)(x2+6x+9)=x3+6x2+9x−3x2−18x−27=x3+3x2−9x−27(x-3)\left(x^2+6x+9\right) = x^3 + 6x^2 + 9x - 3x^2 - 18x - 27 = x^3 + 3x^2 - 9x - 27

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics. By following the steps outlined in this article, you can master this skill and apply it to a wide range of problems. Remember to always distribute the terms, multiply the terms, combine like terms, and check for sum or difference of cubes. With practice and patience, you can become proficient in simplifying algebraic expressions and tackle even the most complex problems with confidence.

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Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to distribute the terms. This involves multiplying each term in the first expression by each term in the second expression.

Q: How do I know if an expression is a sum or difference of cubes?

A: To determine if an expression is a sum or difference of cubes, you need to check if it can be written in the form a3+b3a^3 + b^3 or a3−b3a^3 - b^3. If it can be written in this form, then it is a sum or difference of cubes.

Q: What is the FOIL method?

A: The FOIL method is a technique used to multiply two binomials. It stands for "First, Outer, Inner, Last," and it involves multiplying the first terms, then the outer terms, then the inner terms, and finally the last terms.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract terms that have the same variable and exponent. For example, if you have the expression x2+3x2x^2 + 3x^2, you can combine the like terms by adding them together to get 4x24x^2.

Q: What is the difference between a sum and a difference of cubes?

A: A sum of cubes is an expression that can be written in the form a3+b3a^3 + b^3, while a difference of cubes is an expression that can be written in the form a3−b3a^3 - b^3. For example, x3+27x^3 + 27 is a sum of cubes, while x3−27x^3 - 27 is a difference of cubes.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, you need to follow the same steps as you would for a single variable. This involves distributing the terms, multiplying the terms, combining like terms, and checking for sum or difference of cubes.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Forgetting to distribute the terms
  • Not combining like terms
  • Not checking for sum or difference of cubes
  • Making errors when multiplying the terms

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through examples and exercises in a math textbook or online resource. You can also try simplifying expressions on your own and then checking your work to make sure you got the correct answer.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

  • Solving equations and inequalities
  • Graphing functions
  • Modeling real-world situations
  • Making predictions and forecasts

Q: How can I use technology to help me simplify algebraic expressions?

A: You can use technology, such as calculators or computer software, to help you simplify algebraic expressions. For example, you can use a calculator to evaluate expressions and check your work, or you can use computer software to graph functions and visualize the results.

Q: What are some tips for simplifying algebraic expressions quickly and efficiently?

A: Some tips for simplifying algebraic expressions quickly and efficiently include:

  • Following the order of operations
  • Using the FOIL method to multiply binomials
  • Combining like terms as soon as possible
  • Checking for sum or difference of cubes

Q: How can I overcome common obstacles when simplifying algebraic expressions?

A: Some common obstacles to overcome when simplifying algebraic expressions include:

  • Difficulty with distribution and multiplication
  • Trouble combining like terms
  • Struggling with sum and difference of cubes
  • Making errors when simplifying expressions

Q: What are some resources for learning more about simplifying algebraic expressions?

A: Some resources for learning more about simplifying algebraic expressions include:

  • Math textbooks and online resources
  • Video tutorials and online courses
  • Practice problems and exercises
  • Real-world applications and examples