Simplify The Following Expression By Combining Like Terms:${ -4 + 2z^2 - 5z + 8 + Z^2 + 9z }$ { [?]z^2 + \square Z + \square \}

Mar 11, 2025 by ADMIN 132 views

Simplify the Given Expression by Combining Like Terms

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Introduction

In algebra, combining like terms is a fundamental concept that helps simplify complex expressions. It involves adding or subtracting terms that have the same variable and exponent. In this article, we will simplify the given expression by combining like terms.

The Given Expression

The given expression is:

{ -4 + 2z^2 - 5z + 8 + z^2 + 9z \}

Step 1: Identify Like Terms

To simplify the expression, we need to identify like terms. Like terms are terms that have the same variable and exponent. In this case, we have two like terms with the variable $z^2$ , two like terms with the variable $z$ , and no constant terms.

Step 2: Combine Like Terms

Now that we have identified the like terms, we can combine them. We will start by combining the like terms with the variable $z^2$ .

{ 2z^2 + z^2 = 3z^2 \}

Next, we will combine the like terms with the variable $z$ .

{ -5z + 9z = 4z \}

Step 3: Simplify the Expression

Now that we have combined the like terms, we can simplify the expression by adding the constants.

{ -4 + 8 = 4 \}

Step 4: Write the Simplified Expression

The simplified expression is:

{ 3z^2 + 4z + 4 \}

Conclusion

In this article, we simplified the given expression by combining like terms. We identified the like terms, combined them, and simplified the expression. The final answer is $\boxed{3z^2 + 4z + 4}$ .

Example Use Case

Combining like terms is an essential skill in algebra that can be applied to a wide range of problems. For example, consider the expression:

{ 2x^2 + 3x - 4 + x^2 - 2x + 5 \}

To simplify this expression, we would follow the same steps as before:

Identify like terms: $2x^2$ and $x^2$ are like terms, as are $3x$ and $-2x$ .
Combine like terms: $2x^2 + x^2 = 3x^2$ and $3x - 2x = x$ .
Simplify the expression: $-4 + 5 = 1$ .
Write the simplified expression: $3x^2 + x + 1$ .

Tips and Tricks

When combining like terms, make sure to add or subtract the coefficients of the like terms.
If a term has a negative coefficient, make sure to include the negative sign when combining like terms.
When simplifying an expression, make sure to combine like terms first, and then simplify the expression.

Common Mistakes

Failing to identify like terms: This can lead to incorrect simplification of the expression.
Failing to combine like terms: This can lead to incorrect simplification of the expression.
Not including the negative sign when combining like terms: This can lead to incorrect simplification of the expression.

Final Answer

The final answer is $\boxed{3z^2 + 4z + 4}$ .

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Introduction

Combining like terms is a fundamental concept in algebra that can be a bit tricky to understand at first. In this article, we will answer some frequently asked questions (FAQs) on combining like terms to help you better understand this concept.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $2x^2$ and $x^2$ are like terms because they both have the variable $x$ and the exponent $2$ .

Q: How do I identify like terms?

A: To identify like terms, you need to look for terms that have the same variable and exponent. You can do this by comparing the coefficients of the terms. If the coefficients are the same, then the terms are like terms.

Q: What is the order of operations when combining like terms?

A: When combining like terms, you need to follow the order of operations (PEMDAS):

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

Q: Can I combine like terms with different coefficients?

A: Yes, you can combine like terms with different coefficients. For example, $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ and the exponent $2$ . To combine them, you would add their coefficients: $2x^2 + 3x^2 = 5x^2$ .

Q: What if I have a negative coefficient?

A: If you have a negative coefficient, you need to include the negative sign when combining like terms. For example, $-2x^2$ and $3x^2$ are like terms because they both have the variable $x$ and the exponent $2$ . To combine them, you would subtract the coefficients: $-2x^2 + 3x^2 = x^2$ .

Q: Can I combine like terms with different variables?

A: No, you cannot combine like terms with different variables. For example, $2x^2$ and $2y^2$ are not like terms because they have different variables ( $x$ and $y$ ).

Q: What if I have a term with a variable and a constant?

A: If you have a term with a variable and a constant, you can combine it with other terms that have the same variable and exponent. For example, $2x^2 + 3$ is a term with a variable ( $x^2$ ) and a constant ( $3$ ). You can combine it with other terms that have the same variable and exponent, such as $x^2$ .

Q: Can I simplify an expression by combining like terms?

A: Yes, you can simplify an expression by combining like terms. For example, consider the expression $2x^2 + 3x^2 + 4x + 5x$ . You can combine the like terms to get $5x^2 + 9x$ .

Q: What are some common mistakes to avoid when combining like terms?

A: Some common mistakes to avoid when combining like terms include:

Failing to identify like terms
Failing to combine like terms
Not including the negative sign when combining like terms
Combining terms with different variables
Not following the order of operations

Conclusion

Combining like terms is an essential skill in algebra that can be used to simplify complex expressions. By following the steps outlined in this article, you can master the art of combining like terms and become a proficient algebraist.

Example Use Case

Combining like terms is an essential skill in algebra that can be applied to a wide range of problems. For example, consider the expression $2x^2 + 3x^2 + 4x + 5x$ . You can combine the like terms to get $5x^2 + 9x$ .

Tips and Tricks

When combining like terms, make sure to add or subtract the coefficients of the like terms.
If a term has a negative coefficient, make sure to include the negative sign when combining like terms.
When simplifying an expression, make sure to combine like terms first, and then simplify the expression.

Common Mistakes

Failing to identify like terms: This can lead to incorrect simplification of the expression.
Failing to combine like terms: This can lead to incorrect simplification of the expression.
Not including the negative sign when combining like terms: This can lead to incorrect simplification of the expression.

Final Answer

The final answer is $\boxed{5x^2 + 9x}$ .