Which Expression Is Equivalent To \left(3 X^5+8 X^3\right)+\left(7 X^2-6\right ]?A. − 4 X 3 + 14 -4 X^3+14 − 4 X 3 + 14 B. − 4 X 5 + 14 X 3 -4 X^5+14 X^3 − 4 X 5 + 14 X 3 C. 3 X 5 + 14 X 3 + 7 X 2 3 X^5+14 X^3+7 X^2 3 X 5 + 14 X 3 + 7 X 2 D. 3 X 5 + 2 X 3 + 7 X 2 3 X^5+2 X^3+7 X^2 3 X 5 + 2 X 3 + 7 X 2 Which Graph Best Represents The Solution

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students to master. In this article, we will explore the process of simplifying algebraic expressions, with a focus on combining like terms and applying the order of operations. We will also examine a specific problem involving the simplification of an algebraic expression and provide step-by-step solutions.

What are Algebraic Expressions?

An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. Variables are letters or symbols that represent unknown values, while constants are numbers that do not change value. Algebraic expressions can be simple, such as 2x, or complex, such as 3x^2 + 4x - 5.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and applying the order of operations. Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.

Step 1: Combine Like Terms

To simplify an algebraic expression, we need to combine like terms. This involves adding or subtracting the coefficients of like terms. For example, if we have the expression 2x + 4x, we can combine the like terms by adding the coefficients: 2x + 4x = 6x.

Step 2: Apply the Order of Operations

The order of operations is a set of rules that dictates the order in which mathematical operations should be performed. The order of operations is as follows:

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.

Simplifying the Given Expression

Now that we have a basic understanding of how to simplify algebraic expressions, let's apply this knowledge to the given problem:

$\left(3 x^5+8 x^3\right)+\left(7 x^2-6\right)$

To simplify this expression, we need to combine like terms and apply the order of operations.

Step 1: Distribute the Negative Sign

The first step is to distribute the negative sign to the terms inside the second set of parentheses:

$\left(3 x^5+8 x^3\right)+\left(-7 x^2+6\right)$

Step 2: Combine Like Terms

Next, we need to combine like terms. We can do this by adding or subtracting the coefficients of like terms:

$3 x^5+8 x^3-7 x^2+6$

Step 3: Apply the Order of Operations

Now that we have combined like terms, we need to apply the order of operations. We can do this by following the order of operations:

1. Parentheses: There are no expressions inside parentheses to evaluate.
2. Exponents: There are no exponential expressions to evaluate.
3. Multiplication and Division: There are no multiplication or division operations to evaluate.
4. Addition and Subtraction: Finally, we can evaluate the addition and subtraction operations from left to right:

$3 x^5+8 x^3-7 x^2+6$

The Final Answer

After simplifying the expression, we get:

$3 x^5+8 x^3-7 x^2+6$

This expression is equivalent to the original expression.

Which Graph Best Represents the Solution?

To determine which graph best represents the solution, we need to consider the characteristics of the simplified expression. The expression $3 x^5+8 x^3-7 x^2+6$ is a polynomial expression with a degree of 5. This means that the graph of the expression will be a curve that opens upwards or downwards.

Conclusion

In this article, we explored the process of simplifying algebraic expressions, with a focus on combining like terms and applying the order of operations. We also examined a specific problem involving the simplification of an algebraic expression and provided step-by-step solutions. By following the steps outlined in this article, students can develop a deeper understanding of algebraic expressions and improve their ability to simplify complex expressions.

Final Answer

The final answer is:

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Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on combining like terms and applying the order of operations. In this article, we will provide a Q&A guide to help students better understand the concepts and techniques involved in simplifying algebraic expressions.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. Variables are letters or symbols that represent unknown values, while constants are numbers that do not change value.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictates the order in which mathematical operations should be performed. The order of operations is as follows:

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 combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of like terms. For example, if you have the expression 2x + 4x, you can combine the like terms by adding the coefficients: 2x + 4x = 6x.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents an unknown value, while a constant is a number that does not change value.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to combine like terms and apply the order of operations. Here are the steps to follow:

1. Distribute the negative sign to the terms inside the second set of parentheses.
2. Combine like terms by adding or subtracting the coefficients of like terms.
3. Apply the order of operations by following the order of operations.

Q: What is the final answer to the given problem?

A: The final answer to the given problem is:

C. $3 x^5+14 x^3+7 x^2$

Q: Which graph best represents the solution?

A: The graph that best represents the solution is a curve that opens upwards or downwards, as the expression $3 x^5+8 x^3-7 x^2+6$ is a polynomial expression with a degree of 5.

Q: Can you provide more examples of simplifying algebraic expressions?

A: Here are a few more examples of simplifying algebraic expressions:

• Simplify the expression: $2x^2 + 3x - 4x^2$
• Simplify the expression: $x^3 + 2x^2 - 3x^3$
• Simplify the expression: $4x^2 + 2x - 3x^2$

Answer Key

• Simplify the expression: $2x^2 + 3x - 4x^2 = -2x^2 + 3x$
• Simplify the expression: $x^3 + 2x^2 - 3x^3 = -2x^3 + 2x^2$
• Simplify the expression: $4x^2 + 2x - 3x^2 = x^2 + 2x$

Conclusion

In this article, we provided a Q&A guide to help students better understand the concepts and techniques involved in simplifying algebraic expressions. We also provided examples of simplifying algebraic expressions and answered common questions about the topic. By following the steps outlined in this article, students can develop a deeper understanding of algebraic expressions and improve their ability to simplify complex expressions.