What Is The Product Of \[$(d-9)\$\] And \[$(2d^2 + 11d - 4)\$\]?A. \[$2d^3 + 7d^2 - 95d + 36\$\]B. \[$2d^3 - 7d^2 - 103d + 36\$\]C. \[$2d^3 - 7d^2 - 95d + 36\$\]
In algebra, the product of two expressions is obtained by multiplying each term of the first expression by each term of the second expression. This process involves the use of the distributive property, which states that for any real numbers a, b, and c:
a(b + c) = ab + ac
In this article, we will explore the product of two algebraic expressions, {(d-9)$}$ and {(2d^2 + 11d - 4)$}$, and determine the correct answer among the given options.
Understanding the Distributive Property
To find the product of the two expressions, we will apply the distributive property. This involves multiplying each term of the first expression by each term of the second expression.
Step 1: Multiply the First Term of the First Expression by Each Term of the Second Expression
The first term of the first expression is {d$}$. We will multiply this term by each term of the second expression:
- {d \times 2d^2 = 2d^3$}$
- {d \times 11d = 11d^2$}$
- {d \times -4 = -4d$}$
Step 2: Multiply the Second Term of the First Expression by Each Term of the Second Expression
The second term of the first expression is {-9$}$. We will multiply this term by each term of the second expression:
- {-9 \times 2d^2 = -18d^2$}$
- {-9 \times 11d = -99d$}$
- {-9 \times -4 = 36$}$
Step 3: Combine Like Terms
Now that we have multiplied each term of the first expression by each term of the second expression, we can combine like terms:
- ${2d^3 + 11d^2 - 4d - 18d^2 - 99d + 36\$}
- ${2d^3 - 7d^2 - 103d + 36\$}
Conclusion
Based on the steps outlined above, we can see that the product of the two expressions {(d-9)$}$ and {(2d^2 + 11d - 4)$}$ is ${2d^3 - 7d^2 - 103d + 36\$}. This matches option B.
Answer
The correct answer is:
- B. ${2d^3 - 7d^2 - 103d + 36\$}
Why is this Important?
Understanding how to find the product of two algebraic expressions is an essential skill in algebra and mathematics. It allows us to simplify complex expressions and solve equations. In this article, we have demonstrated how to apply the distributive property to find the product of two expressions.
Real-World Applications
The concept of finding the product of two algebraic expressions has numerous real-world applications. For example, in physics, the product of two expressions can be used to calculate the force exerted on an object. In engineering, it can be used to determine the stress on a material.
Common Mistakes
When finding the product of two algebraic expressions, it is essential to apply the distributive property correctly. Some common mistakes include:
- Failing to distribute each term of the first expression to each term of the second expression
- Not combining like terms correctly
- Not checking the answer for accuracy
Tips and Tricks
To find the product of two algebraic expressions quickly and accurately, follow these tips and tricks:
- Use the distributive property to multiply each term of the first expression by each term of the second expression
- Combine like terms carefully
- Check the answer for accuracy by plugging in values for the variables
Conclusion
In this article, we will address some of the most frequently asked questions about finding the product of two algebraic expressions.
Q: What is the distributive property?
A: The distributive property is a mathematical concept that allows us to multiply each term of one expression by each term of another expression. It states that for any real numbers a, b, and c:
a(b + c) = ab + ac
Q: How do I apply the distributive property to find the product of two expressions?
A: To apply the distributive property, you need to multiply each term of the first expression by each term of the second expression. This involves breaking down each expression into its individual terms and then multiplying each term of the first expression by each term of the second expression.
Q: What is the difference between multiplying two expressions and finding the product of two expressions?
A: Multiplying two expressions involves multiplying each term of one expression by each term of the other expression, but it does not involve combining like terms. Finding the product of two expressions, on the other hand, involves multiplying each term of one expression by each term of the other expression and then combining like terms.
Q: How do I combine like terms?
A: To combine like terms, you need to identify the terms that have the same variable and exponent, and then add or subtract their coefficients. For example, if you have the terms 2x and 3x, you can combine them by adding their coefficients to get 5x.
Q: What are some common mistakes to avoid when finding the product of two expressions?
A: Some common mistakes to avoid when finding the product of two expressions include:
- Failing to distribute each term of the first expression to each term of the second expression
- Not combining like terms correctly
- Not checking the answer for accuracy
Q: How can I check the answer for accuracy?
A: To check the answer for accuracy, you can plug in values for the variables and see if the result is correct. You can also use a calculator or a computer algebra system to check the answer.
Q: What are some real-world applications of finding the product of two expressions?
A: Finding the product of two expressions has numerous real-world applications, including:
- Calculating the force exerted on an object in physics
- Determining the stress on a material in engineering
- Solving equations in mathematics
Q: How can I practice finding the product of two expressions?
A: You can practice finding the product of two expressions by working through examples and exercises in a textbook or online resource. You can also try solving problems on your own and then checking your answers with a calculator or a computer algebra system.
Q: What are some tips and tricks for finding the product of two expressions quickly and accurately?
A: Some tips and tricks for finding the product of two expressions quickly and accurately include:
- Using the distributive property to multiply each term of the first expression by each term of the second expression
- Combining like terms carefully
- Checking the answer for accuracy by plugging in values for the variables
Conclusion
In conclusion, finding the product of two algebraic expressions is an essential skill in algebra and mathematics. By applying the distributive property and combining like terms correctly, we can simplify complex expressions and solve equations. In this article, we have addressed some of the most frequently asked questions about finding the product of two expressions and provided tips and tricks for finding the product quickly and accurately.