Simplify The Expression: ${ \left(-p^2 + 4p - 3\right)\left(p^2 + 2\right) = \square }$

Mar 13, 2025 by ADMIN 89 views

Simplify the Expression: A Comprehensive Guide to Algebraic Manipulation

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will delve into the world of algebra and explore the process of simplifying a given expression. We will use the expression $\left(-p^2 + 4p - 3\right)\left(p^2 + 2\right)$ as a case study and demonstrate the step-by-step process of simplifying it.

Understanding the Expression

Before we begin simplifying the expression, it is essential to understand its structure and components. The given expression is a product of two binomials, which are $\left(-p^2 + 4p - 3\right)$ and $\left(p^2 + 2\right)$ . To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ .

Applying the Distributive Property

To simplify the expression, we need to apply the distributive property to each term in the first binomial. This means that we need to multiply each term in the first binomial by each term in the second binomial. The distributive property can be applied as follows:

$\left(-p^2 + 4p - 3\right)\left(p^2 + 2\right) = \left(-p^2\right)\left(p^2 + 2\right) + \left(4p\right)\left(p^2 + 2\right) + \left(-3\right)\left(p^2 + 2\right)$

Expanding the Terms

Now that we have applied the distributive property, we need to expand each term. This involves multiplying each term in the first binomial by each term in the second binomial. The expanded terms are as follows:

$\left(-p^2\right)\left(p^2 + 2\right) = -p^4 - 2p^2$

$\left(4p\right)\left(p^2 + 2\right) = 4p^3 + 8p$

$\left(-3\right)\left(p^2 + 2\right) = -3p^2 - 6$

Combining Like Terms

Now that we have expanded each term, we need to combine like terms. This involves adding or subtracting terms that have the same variable and exponent. The combined terms are as follows:

$-p^4 - 2p^2 + 4p^3 + 8p - 3p^2 - 6$

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression by combining the terms with the same variable and exponent. The simplified expression is as follows:

$-p^4 + 4p^3 - 5p^2 + 8p - 6$

Conclusion

In this article, we have demonstrated the process of simplifying a given expression using the distributive property and combining like terms. We have used the expression $\left(-p^2 + 4p - 3\right)\left(p^2 + 2\right)$ as a case study and shown the step-by-step process of simplifying it. By following these steps, you can simplify any expression and gain a deeper understanding of algebraic manipulation.

Frequently Asked Questions

What is the distributive property? The distributive property is a mathematical concept that states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ .
How do I apply the distributive property? To apply the distributive property, you need to multiply each term in the first binomial by each term in the second binomial.
What is the difference between like terms and unlike terms? Like terms are terms that have the same variable and exponent, while unlike terms are terms that have different variables or exponents.

Tips and Tricks

Use the distributive property to simplify expressions The distributive property is a powerful tool that can be used to simplify expressions.
Combine like terms to simplify expressions Combining like terms is an essential step in simplifying expressions.
Use algebraic manipulation to solve equations Algebraic manipulation is a crucial skill that can be used to solve equations and gain a deeper understanding of mathematics.

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and understanding the concepts and techniques involved can be challenging. In this article, we will address some of the most frequently asked questions about algebraic manipulation, including the distributive property, combining like terms, and solving equations.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . This means that you can distribute a single term to multiple terms inside a set of parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term in the first binomial by each term in the second binomial. For example, if you have the expression $(a + b)(c + d)$ , you would multiply each term in the first binomial by each term in the second binomial to get $ac + ad + bc + bd$ .

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and exponent, while unlike terms are terms that have different variables or exponents. For example, $2x^2$ and $3x^2$ are like terms, while $2x^2$ and $3y^2$ are unlike terms.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the expression $2x^2 + 3x^2$ , you would combine the like terms to get $5x^2$ .

Q: What is the order of operations in algebraic manipulation?

A: The order of operations in algebraic manipulation is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions 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: How do I solve equations using algebraic manipulation?

A: To solve equations using algebraic manipulation, you need to isolate the variable on one side of the equation. This can involve using inverse operations, such as addition and subtraction, multiplication and division, and exponentiation and logarithms.

Q: What are some common algebraic manipulation techniques?

A: Some common algebraic manipulation techniques include:

Factoring: Breaking down an expression into simpler factors.
Simplifying: Combining like terms and eliminating any unnecessary operations.
Expanding: Multiplying out an expression to get a more detailed view of its components.
Canceling: Eliminating common factors between two expressions.

Tips and Tricks

Use the distributive property to simplify expressions The distributive property is a powerful tool that can be used to simplify expressions.
Combine like terms to simplify expressions Combining like terms is an essential step in simplifying expressions.
Use algebraic manipulation to solve equations Algebraic manipulation is a crucial skill that can be used to solve equations and gain a deeper understanding of mathematics.

Conclusion

Algebraic manipulation is a crucial aspect of mathematics, and understanding the concepts and techniques involved can be challenging. By following the tips and tricks outlined in this article, you can improve your skills in algebraic manipulation and gain a deeper understanding of mathematics.

Simplify The Expression: ${ \left(-p^2 + 4p - 3\right)\left(p^2 + 2\right) = \square }$

Introduction

Understanding the Expression

Applying the Distributive Property

Expanding the Terms

Combining Like Terms

Simplifying the Expression

Conclusion

Frequently Asked Questions

Tips and Tricks

Further Reading

Introduction

Q&A

Q: What is the distributive property?

Q: How do I apply the distributive property?

Q: What is the difference between like terms and unlike terms?

Q: How do I combine like terms?

Q: What is the order of operations in algebraic manipulation?

Q: How do I solve equations using algebraic manipulation?

Q: What are some common algebraic manipulation techniques?

Tips and Tricks

Further Reading

Conclusion