Forgetting to distribute the terms

Oct 13, 2025 by talynt.org 142 views

Understanding the Problem

To find the product of the given expressions, we need to apply the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. This property allows us to multiply each term in the first expression by each term in the second expression.

Step 1: Multiply Each Term in the First Expression by Each Term in the Second Expression

We will start by multiplying the first term in the first expression, 2p, by each term in the second expression, 3p^2, 4p, and -3.

Multiplying 2p by 3p^2

When we multiply 2p by 3p^2, we get 6p^3.

Multiplying 2p by 4p

When we multiply 2p by 4p, we get 8p^2.

Multiplying 2p by -3

When we multiply 2p by -3, we get -6p.

Multiplying 7 by 3p^2

When we multiply 7 by 3p^2, we get 21p^2.

Multiplying 7 by 4p

When we multiply 7 by 4p, we get 28p.

Multiplying 7 by -3

When we multiply 7 by -3, we get -21.

Step 2: Combine Like Terms

Now that we have multiplied each term in the first expression by each term in the second expression, we can combine like terms to simplify the expression.

Combining 6p^3 and 0p^3

There are no like terms to combine for 6p^3.

Combining 8p^2 and 21p^2

When we combine 8p^2 and 21p^2, we get 29p^2.

Combining -6p and 28p

When we combine -6p and 28p, we get 22p.

Combining -21 and 0

There are no like terms to combine for -21.

Step 3: Write the Final Expression

Now that we have combined like terms, we can write the final expression.

The final expression is 6p^3 + 29p^2 + 22p - 21.

Conclusion

The product of ${$ (2p + 7)(3p^2 + 4p - 3)$}$ is 6p^3 + 29p^2 + 22p - 21.

Answer

The correct answer is C. ${ $6p^3 + 29p^2 + 22p - 21\$}$

Discussion

This problem requires the application of the distributive property and the combination of like terms. It is a fundamental concept in algebra and is essential for solving more complex problems in mathematics.

Importance of the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply each term in one expression by each term in another expression. It is a powerful tool for simplifying complex expressions and solving equations.

Real-World Applications

The distributive property has many real-world applications, including:

Science: The distributive property is used in science to simplify complex equations and solve problems in physics, chemistry, and biology.
Engineering: The distributive property is used in engineering to design and optimize complex systems, such as bridges, buildings, and electronic circuits.
Finance: The distributive property is used in finance to simplify complex financial equations and solve problems in accounting, economics, and finance.

Conclusion

In conclusion, the product of ${$ (2p + 7)(3p^2 + 4p - 3)$}$ is 6p^3 + 29p^2 + 22p - 21. The distributive property is a fundamental concept in algebra that allows us to multiply each term in one expression by each term in another expression. It is a powerful tool for simplifying complex expressions and solving equations, and has many real-world applications in science, engineering, and finance.

Q: What is the product of ${$ (2p + 7)(3p^2 + 4p - 3)$}$?

A: The product of ${$ (2p + 7)(3p^2 + 4p - 3)$}$ is 6p^3 + 29p^2 + 22p - 21.

Q: How do I find the product of two expressions?

A: To find the product of two expressions, you need to apply the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. This means that you need to multiply each term in the first expression by each term in the second expression.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to multiply each term in one expression by each term in another expression. It is a powerful tool for simplifying complex expressions and solving equations.

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 2x + 3x, you can combine the like terms by adding the coefficients, which gives you 5x.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, 2x and 3x are like terms because they both have the variable x and the same exponent (which is 1).

Q: Why is it important to simplify expressions?

A: Simplifying expressions is important because it makes it easier to solve equations and understand the relationships between variables. Simplifying expressions also helps to eliminate unnecessary complexity and makes it easier to identify patterns and relationships.

Q: How do I know if an expression is simplified?

A: An expression is simplified if it cannot be further simplified by combining like terms or applying other algebraic properties. In other words, an expression is simplified if it is in its simplest form.

Q: What are some real-world applications of the distributive property?

A: The distributive property has many real-world applications, including:

Science: The distributive property is used in science to simplify complex equations and solve problems in physics, chemistry, and biology.
Engineering: The distributive property is used in engineering to design and optimize complex systems, such as bridges, buildings, and electronic circuits.
Finance: The distributive property is used in finance to simplify complex financial equations and solve problems in accounting, economics, and finance.

Q: How can I practice using the distributive property?

A: You can practice using the distributive property by working through algebraic exercises and problems. You can also try simplifying complex expressions and solving equations using the distributive property.

Q: What are some common mistakes to avoid when using the distributive property?

A: Some common mistakes to avoid when using the distributive property include:

Forgetting to distribute the terms: Make sure to distribute each term in the first expression to each term in the second expression.
Not combining like terms: Make sure to combine like terms to simplify the expression.
Making errors when multiplying: Make sure to multiply each term correctly and avoid making errors when multiplying.

Q: How can I use the distributive property to solve equations?

A: You can use the distributive property to solve equations by simplifying complex expressions and applying algebraic properties to solve for the variables.