Simplify The Expression \[$-3(x+3)^2-3+3x\$\]. What Is The Simplified Expression In Standard Form?A. \[$-3x^2-18x-27\$\] B. \[$-3x^2-15x-30\$\] C. \[$-3x^2+3x+6\$\] D. \[$-3x^2+3x-30\$\]

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Understanding the Problem

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will focus on simplifying the given expression −3(x+3)2−3+3x{-3(x+3)^2-3+3x}. Our goal is to rewrite this expression in standard form, which is a polynomial expression with the variables and constants arranged in a specific order.

The Rules of Simplifying Algebraic Expressions

Before we dive into the solution, let's review the rules of simplifying algebraic expressions:

  • Distributive Property: When multiplying a term by a binomial, we need to multiply each term in the binomial by the term.
  • Combining Like Terms: We can combine like terms by adding or subtracting their coefficients.
  • Order of Operations: We need to follow the order of operations (PEMDAS) when simplifying expressions: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Step 1: Expand the Binomial

The first step is to expand the binomial (x+3)2{(x+3)^2}. To do this, we need to multiply each term in the binomial by the other term.

−3(x+3)2−3+3x{-3(x+3)^2-3+3x}

Using the distributive property, we get:

−3(x2+6x+9)−3+3x{-3(x^2+6x+9)-3+3x}

Step 2: Multiply the Terms

Next, we need to multiply the terms inside the parentheses by the coefficient -3.

−3x2−18x−27−3+3x{-3x^2-18x-27-3+3x}

Step 3: Combine Like Terms

Now, we can combine like terms by adding or subtracting their coefficients.

−3x2−18x−27−3+3x{-3x^2-18x-27-3+3x}

Combining the like terms, we get:

−3x2−15x−30{-3x^2-15x-30}

The Final Answer

Therefore, the simplified expression in standard form is:

−3x2−15x−30{-3x^2-15x-30}

This is the correct answer among the options provided.

Conclusion

Frequently Asked Questions

In this article, we will address some of the most common questions related to simplifying algebraic expressions.

Q: What is the distributive property in algebra?

A: The distributive property is a rule in algebra that allows us to multiply a term by a binomial. It states that when multiplying a term by a binomial, we need to multiply each term in the binomial by the term.

Q: How do I combine like terms in an algebraic expression?

A: To combine like terms, we need to add or subtract their coefficients. Like terms are terms that have the same variable raised to the same power.

Q: What is the order of operations in algebra?

A: The order of operations in algebra is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an algebraic expression with exponents?

A: To simplify an algebraic expression with exponents, we need to follow the order of operations. We need to evaluate any exponential expressions first, and then perform any multiplication and division operations.

Q: What is the difference between a polynomial and an algebraic expression?

A: A polynomial is a type of algebraic expression that consists of variables and constants raised to non-negative integer powers. An algebraic expression, on the other hand, can be any expression that involves variables and constants.

Q: How do I simplify an algebraic expression with fractions?

A: To simplify an algebraic expression with fractions, we need to follow the order of operations. We need to evaluate any exponential expressions first, and then perform any multiplication and division operations.

Q: What is the final answer to the expression −3(x+3)2−3+3x{-3(x+3)^2-3+3x}?

A: The final answer to the expression −3(x+3)2−3+3x{-3(x+3)^2-3+3x} is −3x2−15x−30{-3x^2-15x-30}.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. By following the distributive property, combining like terms, and following the order of operations, we can simplify complex expressions and rewrite them in standard form. In this article, we addressed some of the most common questions related to simplifying algebraic expressions and provided a step-by-step guide to simplifying the expression −3(x+3)2−3+3x{-3(x+3)^2-3+3x}.