Which Expression Is Equivalent To \[$8p^3 + 27\$\]?

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Introduction

In mathematics, algebraic expressions are used to represent various mathematical operations and relationships. One of the fundamental concepts in algebra is the concept of equivalent expressions, which are expressions that have the same value or result when evaluated. In this article, we will explore the concept of equivalent expressions and determine which expression is equivalent to ${8p^3 + 27\$}.

Understanding the Concept of Equivalent Expressions

Equivalent expressions are expressions that have the same value or result when evaluated. They can be obtained by applying various mathematical operations such as addition, subtraction, multiplication, and division, as well as using algebraic identities and formulas. Equivalent expressions are useful in simplifying complex expressions, solving equations, and proving mathematical theorems.

The Concept of Cubic Expressions

Cubic expressions are expressions that contain a variable raised to the power of 3. They are commonly used in algebra and geometry to represent the volume of a cube or the surface area of a cube. Cubic expressions can be written in the form ax3+bx2+cx+d{ax^3 + bx^2 + cx + d}, where a{a}, b{b}, c{c}, and d{d} are constants, and x{x} is the variable.

The Expression ${8p^3 + 27\$}

The expression ${8p^3 + 27\$} is a cubic expression that contains a variable p{p} raised to the power of 3. The expression can be written in the form 8p3+27{8p^3 + 27}, where 8{8} and 27{27} are constants.

Factoring the Expression ${8p^3 + 27\$}

To determine which expression is equivalent to ${8p^3 + 27\$}, we need to factor the expression. Factoring an expression involves expressing it as a product of simpler expressions. In this case, we can factor the expression ${8p^3 + 27\$} as follows:

${8p^3 + 27 = (2p)^3 + 3^3}

Applying the Sum of Cubes Formula

The expression (2p)3+33{(2p)^3 + 3^3} can be simplified using the sum of cubes formula, which states that:

(a+b)3=a3+3a2b+3ab2+b3{(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3}

Applying this formula to the expression (2p)3+33{(2p)^3 + 3^3}, we get:

(2p)3+33=(2p)3+33=(2p+3)((2p)2−(2p)(3)+32){(2p)^3 + 3^3 = (2p)^3 + 3^3 = (2p + 3)((2p)^2 - (2p)(3) + 3^2)}

Simplifying the Expression

Simplifying the expression further, we get:

(2p+3)((2p)2−(2p)(3)+32)=(2p+3)(4p2−6p+9){(2p + 3)((2p)^2 - (2p)(3) + 3^2) = (2p + 3)(4p^2 - 6p + 9)}

Conclusion

In conclusion, the expression ${8p^3 + 27\$} is equivalent to the expression (2p+3)(4p2−6p+9){(2p + 3)(4p^2 - 6p + 9)}. This expression can be obtained by factoring the original expression and applying the sum of cubes formula.

Final Answer

The final answer is: (2p + 3)(4p^2 - 6p + 9)

Introduction

In our previous article, we explored the concept of equivalent expressions and determined which expression is equivalent to ${8p^3 + 27\$}. In this article, we will answer some frequently asked questions related to equivalent expressions and cubic expressions.

Q: What is the difference between equivalent expressions and similar expressions?

A: Equivalent expressions are expressions that have the same value or result when evaluated, while similar expressions are expressions that have the same form or structure but may not have the same value or result.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you can use various methods such as factoring, simplifying, and applying algebraic identities and formulas.

Q: What is the sum of cubes formula?

A: The sum of cubes formula is a mathematical formula that states:

(a+b)3=a3+3a2b+3ab2+b3{(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3}

This formula can be used to simplify expressions that contain the sum of cubes.

Q: How do I factor a cubic expression?

A: To factor a cubic expression, you can use various methods such as factoring out the greatest common factor, using the sum of cubes formula, or applying other algebraic identities and formulas.

Q: What is the difference between a cubic expression and a quadratic expression?

A: A cubic expression is an expression that contains a variable raised to the power of 3, while a quadratic expression is an expression that contains a variable raised to the power of 2.

Q: How do I simplify a cubic expression?

A: To simplify a cubic expression, you can use various methods such as factoring, simplifying, and applying algebraic identities and formulas.

Q: What is the significance of equivalent expressions in mathematics?

A: Equivalent expressions are significant in mathematics because they can be used to simplify complex expressions, solve equations, and prove mathematical theorems.

Q: How do I apply the sum of cubes formula to a given expression?

A: To apply the sum of cubes formula to a given expression, you can identify the sum of cubes pattern and then use the formula to simplify the expression.

Q: What are some common algebraic identities and formulas that can be used to simplify expressions?

A: Some common algebraic identities and formulas that can be used to simplify expressions include the sum of cubes formula, the difference of cubes formula, and the Pythagorean identity.

Q: How do I determine if an expression is a cubic expression?

A: To determine if an expression is a cubic expression, you can look for the variable raised to the power of 3.

Q: What is the difference between a cubic expression and a polynomial expression?

A: A cubic expression is a type of polynomial expression that contains a variable raised to the power of 3, while a polynomial expression is a general term that refers to an expression that contains variables raised to various powers.

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, you can use various methods such as factoring, simplifying, and applying algebraic identities and formulas.

Q: What is the significance of polynomial expressions in mathematics?

A: Polynomial expressions are significant in mathematics because they can be used to model real-world problems, solve equations, and prove mathematical theorems.

Conclusion

In conclusion, equivalent expressions and cubic expressions are fundamental concepts in mathematics that can be used to simplify complex expressions, solve equations, and prove mathematical theorems. By understanding these concepts and applying various algebraic identities and formulas, you can simplify expressions and solve problems with ease.

Final Answer

The final answer is: (2p + 3)(4p^2 - 6p + 9)