What Is The Product Of The Following Expression?${ \left(-2 D^2+s\right)\left(5 D^2-6 S\right) }$A. { -10 D^4+17 D^2 S-6 S^2$}$B. { -10 D^4+17 D^4 S^2-6 S^2$}$C. { -10 D^4-7 D^2 S-6 S^2$} D . \[ D. \[ D . \[ -10 D^4+17
In this article, we will explore the concept of multiplying two algebraic expressions and find the product of the given expression. We will use the distributive property to simplify the expression and find the final result.
Understanding the Distributive Property
The distributive property is a fundamental concept in algebra that allows us to multiply two or more expressions by distributing each term of one expression to each term of the other expression. This property is represented by the following equation:
a(b + c) = ab + ac
where a, b, and c are algebraic expressions.
Applying the Distributive Property
To find the product of the given expression, we will apply the distributive property to each term of the first expression and multiply it by each term of the second expression.
The given expression is:
(-2d^2 + s)(5d^2 - 6s)
Using the distributive property, we can rewrite this expression as:
(-2d2)(5d2) + (-2d^2)(-6s) + s(5d^2) + s(-6s)
Simplifying the Expression
Now, we will simplify each term of the expression by multiplying the coefficients and variables.
(-2d2)(5d2) = -10d^4
(-2d^2)(-6s) = 12d^2s
s(5d^2) = 5d^2s
s(-6s) = -6s^2
Combining Like Terms
Now, we will combine like terms by adding or subtracting the coefficients of the same variables.
-10d^4 + 12d^2s + 5d^2s - 6s^2
Combine like terms:
-10d^4 + 17d^2s - 6s^2
Conclusion
The product of the given expression is:
-10d^4 + 17d^2s - 6s^2
This is the correct answer among the options provided.
Comparison with Options
Let's compare our result with the options provided:
A. -10d^4 + 17d4s2 - 6s^2
B. -10d^4 + 17d4s2 - 6s^2
C. -10d^4 - 7d^2s - 6s^2
D. -10d^4 + 17d^2s - 6s^2
Our result matches option D.
Final Answer
The final answer is:
In this article, we will address some of the most frequently asked questions about the product of the given expression. We will provide detailed explanations and examples to help you understand the concepts better.
Q: What is the distributive property?
A: The distributive property is a fundamental concept in algebra that allows us to multiply two or more expressions by distributing each term of one expression to each term of the other expression. This property is represented by the following equation:
a(b + c) = ab + ac
where a, b, and c are algebraic expressions.
Q: How do I apply the distributive property to the given expression?
A: To apply the distributive property to the given expression, we need to multiply each term of the first expression by each term of the second expression. This can be done by following the order of operations (PEMDAS):
- Multiply the first term of the first expression by the first term of the second expression.
- Multiply the first term of the first expression by the second term of the second expression.
- Multiply the second term of the first expression by the first term of the second expression.
- Multiply the second term of the first expression by the second term of the second expression.
Q: What is the difference between combining like terms and simplifying an expression?
A: Combining like terms involves adding or subtracting the coefficients of the same variables, while simplifying an expression involves reducing the expression to its simplest form by performing operations such as addition, subtraction, multiplication, and division.
Q: How do I know which terms to combine when simplifying an expression?
A: When simplifying an expression, you need to identify the like terms and combine them by adding or subtracting their coefficients. Like terms are terms that have the same variable(s) raised to the same power.
Q: What is the final answer to the given expression?
A: The final answer to the given expression is:
-10d^4 + 17d^2s - 6s^2
Q: Why is it important to simplify expressions?
A: Simplifying expressions is important because it helps to:
- Reduce the complexity of the expression
- Make it easier to understand and work with
- Avoid errors and mistakes
- Improve the accuracy of calculations
Q: Can I use the distributive property to simplify expressions with more than two terms?
A: Yes, you can use the distributive property to simplify expressions with more than two terms. Simply apply the distributive property to each term of the first expression and multiply it by each term of the second expression.
Q: What are some common mistakes to avoid when simplifying expressions?
A: Some common mistakes to avoid when simplifying expressions include:
- Forgetting to combine like terms
- Not following the order of operations (PEMDAS)
- Making errors when multiplying or dividing expressions
- Not checking the final answer for accuracy
Conclusion
In this article, we have addressed some of the most frequently asked questions about the product of the given expression. We have provided detailed explanations and examples to help you understand the concepts better. Remember to always follow the order of operations (PEMDAS) and combine like terms when simplifying expressions.