Find The Difference: ( 4 X 2 + 8 X + 2 ) − ( 2 X + 3 \left(4x^2+8x+2\right)-(2x+3 ( 4 X 2 + 8 X + 2 ) − ( 2 X + 3 ]A. 7 X 2 − 3 X + 1 7x^2-3x+1 7 X 2 − 3 X + 1 B. 6 X 2 + 8 X + 3 6x^2+8x+3 6 X 2 + 8 X + 3 C. 4 X 2 + 6 X − 1 4x^2+6x-1 4 X 2 + 6 X − 1 D. 14 X 2 + 3 X − 3 14x^2+3x-3 14 X 2 + 3 X − 3

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Understanding the Problem

When simplifying algebraic expressions, it's essential to apply the correct order of operations and combine like terms. In this problem, we're given an expression with two terms: (4x2+8x+2)\left(4x^2+8x+2\right) and (2x+3)(2x+3). Our goal is to find the difference between these two expressions.

Step 1: Distribute the Negative Sign

To find the difference, we need to distribute the negative sign to the second term. This means multiplying each term in the second expression by -1.

-(2x+3) = -2x - 3

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. The first expression is (4x2+8x+2)\left(4x^2+8x+2\right), and the second expression is −2x−3-2x - 3. We can combine the like terms by adding or subtracting the coefficients of the same variables.

4x^2 + 8x + 2 - 2x - 3

Step 3: Simplify the Expression

Now, let's simplify the expression by combining the like terms.

4x^2 + (8x - 2x) + (2 - 3)

Step 4: Combine the Coefficients

We can combine the coefficients of the like terms.

4x^2 + 6x - 1

Conclusion

The difference between the two expressions is 4x2+6x−14x^2 + 6x - 1. This is the correct answer.

Comparison with the Options

Let's compare our answer with the options provided.

  • A. 7x2−3x+17x^2-3x+1
  • B. 6x2+8x+36x^2+8x+3
  • C. 4x2+6x−14x^2+6x-1
  • D. 14x2+3x−314x^2+3x-3

Our answer, 4x2+6x−14x^2 + 6x - 1, matches option C.

Final Answer

The final answer is C\boxed{C}.

Importance of Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics. It helps us to:

  • Reduce complexity: Simplifying expressions makes them easier to work with and understand.
  • Identify patterns: By simplifying expressions, we can identify patterns and relationships between variables.
  • Solve equations: Simplifying expressions is essential for solving equations and inequalities.
  • Analyze functions: Simplifying expressions helps us to analyze functions and their behavior.

Tips for Simplifying Algebraic Expressions

Here are some tips for simplifying algebraic expressions:

  • Apply the order of operations: Follow the order of operations (PEMDAS) to simplify expressions.
  • Combine like terms: Combine like terms by adding or subtracting the coefficients of the same variables.
  • Distribute negative signs: Distribute negative signs to the terms in the expression.
  • Simplify fractions: Simplify fractions by dividing the numerator and denominator by their greatest common divisor.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying algebraic expressions:

  • Forgetting to distribute negative signs: Failing to distribute negative signs can lead to incorrect results.
  • Not combining like terms: Failing to combine like terms can make expressions more complex than necessary.
  • Not simplifying fractions: Failing to simplify fractions can make expressions more complex than necessary.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics. By following the correct order of operations, combining like terms, and distributing negative signs, we can simplify expressions and identify patterns and relationships between variables. Remember to avoid common mistakes, such as forgetting to distribute negative signs and not combining like terms. With practice and patience, you can become proficient in simplifying algebraic expressions.

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Frequently Asked Questions

Q: What is the order of operations in simplifying algebraic expressions?

A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I distribute negative signs in simplifying algebraic expressions?

A: To distribute a negative sign, multiply each term in the expression by -1. For example, -(2x+3) = -2x - 3.

Q: What is the difference between combining like terms and distributing negative signs?

A: Combining like terms involves adding or subtracting the coefficients of the same variables, while distributing negative signs involves multiplying each term in the expression by -1.

Q: How do I simplify fractions in algebraic expressions?

A: To simplify fractions, divide the numerator and denominator by their greatest common divisor. For example, 6/8 = 3/4.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is essential for solving equations and inequalities, analyzing functions, and identifying patterns and relationships between variables.

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

A: Some common mistakes to avoid include forgetting to distribute negative signs, not combining like terms, and not simplifying fractions.

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

A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms.

Q: Can you provide an example of simplifying an algebraic expression?

A: Let's simplify the expression 2x^2 + 3x + 2 - x^2 - 2x. First, distribute the negative sign to the second term: 2x^2 + 3x + 2 - x^2 - 2x = 2x^2 + (3x - 2x) + (2 - x^2). Then, combine like terms: 2x^2 + x^2 + x + 2 - x^2 = x^2 + x + 2.

Q: How do I know when to simplify an algebraic expression?

A: You should simplify an algebraic expression when it is necessary to solve an equation or inequality, analyze a function, or identify patterns and relationships between variables.

Q: Can you provide a real-world example of simplifying algebraic expressions?

A: In physics, the equation for the motion of an object under constant acceleration is s = ut + (1/2)at^2, where s is the distance traveled, u is the initial velocity, t is time, and a is acceleration. To simplify this equation, we can combine like terms: s = ut + (1/2)at^2 = ut + (1/2)at^2.

Q: How do I check my work when simplifying algebraic expressions?

A: To check your work, plug the simplified expression back into the original equation and verify that it is true.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics. By following the correct order of operations, combining like terms, and distributing negative signs, we can simplify expressions and identify patterns and relationships between variables. Remember to avoid common mistakes, such as forgetting to distribute negative signs and not combining like terms. With practice and patience, you can become proficient in simplifying algebraic expressions.

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