Find The Difference. \left(7p + 4p^3 - 8\right) - \left(-3p^2 + 2 - 9p\right ]

Mar 10, 2025 by ADMIN 79 views

Find the Difference: Simplifying Algebraic Expressions

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In algebra, simplifying expressions is a crucial step in solving equations and inequalities. It involves combining like terms and eliminating any unnecessary components. In this article, we will explore how to find the difference between two algebraic expressions, specifically the expression $\left(7p + 4p^3 - 8\right) - \left(-3p^2 + 2 - 9p\right)$ .

Understanding the Problem

The given expression involves two sets of parentheses, each containing a combination of variables and constants. To find the difference, we need to simplify the expression by combining like terms and eliminating any unnecessary components.

Like Terms and Constants

What are Like Terms?

Like terms are terms that have the same variable raised to the same power. In the given expression, we have terms with the variable $p$ raised to different powers, such as $p$ , $p^2$ , and $p^3$ . We also have constants, such as $-8$ and $2$ .

Combining Like Terms

To combine like terms, we need to add or subtract the coefficients of the terms with the same variable raised to the same power. For example, in the expression $7p + 4p^3 - 8$ , we have two terms with the variable $p$ , namely $7p$ and $-9p$ . We can combine these terms by adding their coefficients, which gives us $-2p$ .

Simplifying the Expression

Now that we have a better understanding of like terms and constants, let's simplify the given expression.

Step 1: Distribute the Negative Sign

Distributing the Negative Sign

When we subtract a term, we need to distribute the negative sign to all the terms inside the parentheses. In this case, we have the expression $\left(-3p^2 + 2 - 9p\right)$ . To subtract this expression, we need to distribute the negative sign to all the terms inside the parentheses, which gives us $-3p^2 - 2 + 9p$ .

Step 2: Combine Like Terms

Combining Like Terms

Now that we have distributed the negative sign, we can combine like terms. We have terms with the variable $p$ raised to different powers, such as $p$ , $p^2$ , and $p^3$ . We also have constants, such as $-8$ and $2$ . To combine like terms, we need to add or subtract the coefficients of the terms with the same variable raised to the same power.

Step 3: Simplify the Expression

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression. We have the expression $7p + 4p^3 - 8 - 3p^2 - 2 + 9p$ . To simplify this expression, we need to combine the like terms and eliminate any unnecessary components.

Final Answer

After simplifying the expression, we get:

\left(7p + 4p^3 - 8\right) - \left(-3p^2 + 2 - 9p\right) = 4p^3 + 7p + 9p - 3p^2 - 8 - 2 + 8

Simplifying further, we get:

4p^3 + 16p - 3p^2 - 10

Therefore, the final answer is:

4p^3 + 16p - 3p^2 - 10

Conclusion

In this article, we explored how to find the difference between two algebraic expressions. We simplified the expression by combining like terms and eliminating any unnecessary components. We also discussed the importance of understanding like terms and constants in algebra. By following these steps, we can simplify complex algebraic expressions and arrive at the final answer.

Frequently Asked Questions

Q: What are like terms in algebra?

A: Like terms are terms that have the same variable raised to the same power.

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 raised to the same power.

Q: What is the importance of understanding like terms and constants in algebra?

A: Understanding like terms and constants is crucial in simplifying algebraic expressions and solving equations and inequalities.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to combine like terms and eliminate any unnecessary components.

References

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In our previous article, we explored how to find the difference between two algebraic expressions. We simplified the expression by combining like terms and eliminating any unnecessary components. In this article, we will continue to provide a Q&A guide on algebraic expressions, covering topics such as like terms, constants, and simplifying expressions.

Q&A: Algebraic Expressions

Q: What are algebraic expressions?

A: Algebraic expressions are a combination of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division.

Q: What are like terms in algebra?

A: Like terms are terms that have the same variable raised to the same power.

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 raised to the same power.

Q: What is the importance of understanding like terms and constants in algebra?

A: Understanding like terms and constants is crucial in simplifying algebraic expressions and solving equations and inequalities.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to combine like terms and eliminate any unnecessary components.

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

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that remains the same.

Q: How do I evaluate an algebraic expression?

A: To evaluate an algebraic expression, you need to substitute the values of the variables with the given values and perform the mathematical operations.

Q: What is the order of operations in algebra?

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

Q: How do I solve an equation with variables on both sides?

A: To solve an equation with variables on both sides, you need to isolate the variable by performing the inverse operations on both sides of the equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation with a single variable raised to the power of 1, while a quadratic equation is an equation with a single variable raised to the power of 2.

Examples and Solutions

Example 1: Simplifying an Algebraic Expression

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

Solution:

$2x^2 + 3x - 4 - 2x^2 + x + 2$

Combine like terms:

$2x^2 - 2x^2 + 3x + x - 4 + 2$

Simplify:

$4x - 2$

Example 2: Evaluating an Algebraic Expression

Evaluate the expression: $2x^2 + 3x - 4$ when $x = 2$

Solution:

$2x^2 + 3x - 4$

Substitute $x = 2$ :

$2(2)^2 + 3(2) - 4$

Evaluate:

$2(4) + 6 - 4$

Simplify:

$8 + 6 - 4$

Final answer:

$10$

Conclusion

In this article, we provided a Q&A guide on algebraic expressions, covering topics such as like terms, constants, and simplifying expressions. We also provided examples and solutions to help illustrate the concepts. By following these steps and understanding the concepts, you can simplify complex algebraic expressions and solve equations and inequalities.

Frequently Asked Questions

Q: What are some common algebraic expressions?

A: Some common algebraic expressions include linear equations, quadratic equations, and polynomial expressions.

Q: How do I graph an algebraic expression?

A: To graph an algebraic expression, you need to use a graphing calculator or software to visualize the expression.

Q: What is the difference between a function and an expression in algebra?

A: A function is a relation between a set of inputs and a set of possible outputs, while an expression is a combination of variables, constants, and mathematical operations.

Q: How do I solve a system of equations in algebra?

A: To solve a system of equations, you need to use methods such as substitution, elimination, or graphing to find the solution.