Multiply.$5p(7p^2 - 4p + 9$\]Enter Your Answer In The Box:$\square$

$$

Understanding the Problem

When dealing with algebraic expressions, multiplication is a fundamental operation that helps us simplify and manipulate equations. In this problem, we are given the expression $5p(7p^2 - 4p + 9)$ and we need to multiply it. To do this, we will use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Applying the Distributive Property

To multiply the given expression, we will apply the distributive property by multiplying each term inside the parentheses by the term outside the parentheses, which is $5p$. This means we will multiply $5p$ by $7p^2$, $5p$ by $-4p$, and $5p$ by $9$.

Multiplying $5p$ by $7p^2$

When we multiply $5p$ by $7p^2$, we get $35p^3$. This is because when we multiply two variables with the same base, we add their exponents. In this case, $p$ is the variable and its exponent is $1$. So, $5p \cdot 7p^2 = 5 \cdot 7 \cdot p \cdot p^2 = 35p^3$.

Multiplying $5p$ by $-4p$

When we multiply $5p$ by $-4p$, we get $-20p^2$. This is because when we multiply two variables with the same base, we add their exponents. In this case, $p$ is the variable and its exponent is $1$. So, $5p \cdot -4p = 5 \cdot -4 \cdot p \cdot p = -20p^2$.

Multiplying $5p$ by $9$

When we multiply $5p$ by $9$, we get $45p$. This is because when we multiply a variable by a constant, we simply multiply the constant by the variable. In this case, $5p \cdot 9 = 5 \cdot 9 \cdot p = 45p$.

Combining the Terms

Now that we have multiplied each term inside the parentheses by the term outside the parentheses, we can combine the terms to get the final result. The expression $5p(7p^2 - 4p + 9)$ can be rewritten as $35p^3 - 20p^2 + 45p$.

Conclusion

In this problem, we used the distributive property to multiply the expression $5p(7p^2 - 4p + 9)$. We multiplied each term inside the parentheses by the term outside the parentheses and then combined the terms to get the final result. The final answer is $35p^3 - 20p^2 + 45p$.

Example Use Case

The distributive property is a fundamental concept in algebra that has many practical applications. For example, in physics, the distributive property is used to calculate the force exerted on an object by multiple forces. In economics, the distributive property is used to calculate the total cost of producing a product.

Tips and Tricks

• When multiplying expressions, always use the distributive property to simplify the calculation.
• When combining terms, make sure to add or subtract the coefficients correctly.
• When dealing with variables, always remember to multiply the variables with the same base by adding their exponents.

Common Mistakes

• Failing to use the distributive property when multiplying expressions.
• Adding or subtracting coefficients incorrectly when combining terms.
• Forgetting to multiply variables with the same base by adding their exponents.

Real-World Applications

The distributive property has many real-world applications in fields such as physics, economics, and engineering. For example, in physics, the distributive property is used to calculate the force exerted on an object by multiple forces. In economics, the distributive property is used to calculate the total cost of producing a product.

Conclusion

In conclusion, the distributive property is a fundamental concept in algebra that has many practical applications. By understanding and applying the distributive property, we can simplify complex expressions and solve problems in various fields. The final answer is $35p^3 - 20p^2 + 45p$.

$$

Frequently Asked Questions

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This means that we can multiply a single term by multiple terms inside parentheses by multiplying each term individually.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply each term inside the parentheses by the term outside the parentheses. For example, if we have the expression $5p(7p^2 - 4p + 9)$, we would multiply $5p$ by $7p^2$, $5p$ by $-4p$, and $5p$ by $9$.

Q: What is the difference between multiplying variables and multiplying constants?

A: When multiplying variables, we add their exponents. For example, $p \cdot p^2 = p^3$. However, when multiplying constants, we simply multiply the constants together. For example, $5 \cdot 9 = 45$.

Q: How do I combine like terms?

A: To combine like terms, we add or subtract the coefficients of the terms. For example, if we have the expression $3p + 2p$, we would combine the terms by adding the coefficients: $3p + 2p = 5p$.

Q: What is the final answer to the problem $5p(7p^2 - 4p + 9)$?

A: The final answer to the problem $5p(7p^2 - 4p + 9)$ is $35p^3 - 20p^2 + 45p$.

Q: Can I use the distributive property with fractions?

A: Yes, you can use the distributive property with fractions. For example, if we have the expression $\frac{1}{2}(3x + 4)$, we would multiply $\frac{1}{2}$ by $3x$ and $\frac{1}{2}$ by $4$.

Q: Can I use the distributive property with decimals?

A: Yes, you can use the distributive property with decimals. For example, if we have the expression $0.5(3x + 4)$, we would multiply $0.5$ by $3x$ and $0.5$ by $4$.

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:

• Failing to use the distributive property when multiplying expressions.
• Adding or subtracting coefficients incorrectly when combining terms.
• Forgetting to multiply variables with the same base by adding their exponents.

Q: How do I check my work when using the distributive property?

A: To check your work when using the distributive property, simply multiply the terms inside the parentheses by the term outside the parentheses and then combine the terms. You can also use a calculator or a computer algebra system to check your work.

Q: Can I use the distributive property with negative numbers?

A: Yes, you can use the distributive property with negative numbers. For example, if we have the expression $-3(2x + 4)$, we would multiply $-3$ by $2x$ and $-3$ by $4$.

Q: Can I use the distributive property with zero?

A: Yes, you can use the distributive property with zero. For example, if we have the expression $0(2x + 4)$, we would multiply $0$ by $2x$ and $0$ by $4$, resulting in $0$.

Conclusion

In conclusion, the distributive property is a fundamental concept in algebra that has many practical applications. By understanding and applying the distributive property, we can simplify complex expressions and solve problems in various fields. Remember to use the distributive property to multiply expressions, combine like terms, and check your work.