Simplify The Expression By Combining Like Terms:$\[ 0.4x^2 - 3.6 + 4.6x + 1.2x^2 - 4 \\]Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. The Polynomial Can Be Simplified As
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Understanding the Problem
When simplifying an expression by combining like terms, it is essential to identify and group together the terms that have the same variable and exponent. This process helps to reduce the complexity of the expression and makes it easier to work with.
Identifying Like Terms
Like terms are terms that have the same variable and exponent. In the given expression, we can identify the like terms as follows:
- Terms with the variable x^2: 0.4x^2 and 1.2x^2
- Terms with the variable x: 4.6x
- Constant terms: -3.6 and -4
- Constant term: 1.2
Combining Like Terms
To combine like terms, we need to add or subtract the coefficients of the terms with the same variable and exponent.
Combining Terms with the Variable x^2
The terms with the variable x^2 are 0.4x^2 and 1.2x^2. To combine these terms, we add their coefficients:
0.4x^2 + 1.2x^2 = (0.4 + 1.2)x^2 = 1.6x^2
Combining Terms with the Variable x
The term with the variable x is 4.6x. There are no other terms with the variable x, so we cannot combine this term with any other term.
Combining Constant Terms
The constant terms are -3.6 and -4. To combine these terms, we add their coefficients:
-3.6 + (-4) = -3.6 - 4 = -7.6
Combining the Constant Term
The constant term is 1.2. There are no other constant terms, so we cannot combine this term with any other term.
Simplifying the Expression
Now that we have combined the like terms, we can simplify the expression by adding the combined terms:
1.6x^2 + 4.6x - 7.6 + 1.2
Final Answer
The simplified expression is:
1.6x^2 + 4.6x - 6.4
The final answer is:
Conclusion
Simplifying an expression by combining like terms is an essential skill in algebra. By identifying and grouping together the like terms, we can reduce the complexity of the expression and make it easier to work with. In this problem, we combined the like terms and simplified the expression to obtain the final answer.
Frequently Asked Questions
Q: What are like terms?
A: Like terms are terms that have the same variable and exponent.
Q: How do I combine like terms?
A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable and exponent.
Q: What is the final answer?
A: The final answer is 1.6x^2 + 4.6x - 6.4.
Q: Why is simplifying an expression by combining like terms important?
A: Simplifying an expression by combining like terms is important because it reduces the complexity of the expression and makes it easier to work with.
Step-by-Step Solution
- Identify the like terms in the expression.
- Combine the like terms by adding or subtracting their coefficients.
- Simplify the expression by adding the combined terms.
- Write the final answer.
Tips and Tricks
- Make sure to identify all the like terms in the expression.
- Use the correct operation (addition or subtraction) when combining like terms.
- Simplify the expression by adding the combined terms.
- Check your final answer to make sure it is correct.
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Q&A: Simplifying Expressions by Combining Like Terms
Q: What are like terms?
A: Like terms are terms that have the same variable and exponent. For example, 2x and 4x are like terms because they both have the variable x and the same exponent (which is 1).
Q: How do I identify like terms in an expression?
A: To identify like terms, you need to look for terms that have the same variable and exponent. You can do this by:
- Looking for terms with the same variable (e.g., x, y, z)
- Checking if the terms have the same exponent (e.g., x^2, x^3, x^4)
- Combining terms with the same variable and exponent
Q: How do I combine like terms?
A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable and exponent. For example:
- 2x + 4x = (2 + 4)x = 6x
- 3y^2 - 2y^2 = (3 - 2)y^2 = y^2
Q: What is the difference between combining like terms and simplifying an expression?
A: Combining like terms is the process of adding or subtracting the coefficients of terms with the same variable and exponent. Simplifying an expression is the process of reducing an expression to its simplest form by combining like terms and eliminating any unnecessary terms.
Q: Why is simplifying an expression by combining like terms important?
A: Simplifying an expression by combining like terms is important because it:
- Reduces the complexity of the expression
- Makes it easier to work with the expression
- Helps to identify patterns and relationships in the expression
- Makes it easier to solve equations and inequalities
Q: What are some common mistakes to avoid when simplifying expressions by combining like terms?
A: Some common mistakes to avoid when simplifying expressions by combining like terms include:
- Failing to identify all the like terms in the expression
- Using the wrong operation (addition or subtraction) when combining like terms
- Not simplifying the expression to its simplest form
- Not checking the final answer to make sure it is correct
Q: How do I check my work when simplifying expressions by combining like terms?
A: To check your work when simplifying expressions by combining like terms, you can:
- Review the original expression to make sure you identified all the like terms
- Check the simplified expression to make sure it is in its simplest form
- Use a calculator or computer software to verify the simplified expression
- Check the final answer to make sure it is correct
Q: What are some real-world applications of simplifying expressions by combining like terms?
A: Simplifying expressions by combining like terms has many real-world applications, including:
- Solving equations and inequalities in physics and engineering
- Modeling population growth and decay in biology and economics
- Analyzing data and making predictions in statistics and data science
- Solving optimization problems in business and finance
Conclusion
Simplifying expressions by combining like terms is an essential skill in algebra and has many real-world applications. By understanding how to identify and combine like terms, you can reduce the complexity of expressions and make it easier to work with them. Remember to check your work and verify the final answer to make sure it is correct.