What Is The Product Of The Following Expression?${ (-2d^2 + S)(5d^2 - 6s) }$A. { -10d^4 + 17d^2s - 6s^2$}$ B. { -10d^4 - 7d^2s - 6s^2$}$ C. { -10e^4 + 17d^2s + 6s^2$}$

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Understanding the Problem

The given problem involves finding the product of two quadratic expressions. To solve this, we need to apply the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. We will use this property to multiply the two given expressions.

The Given Expression

The given expression is:

(−2d2+s)(5d2−6s)(-2d^2 + s)(5d^2 - 6s)

Step 1: Apply the Distributive Property

To find the product, we will apply the distributive property by multiplying each term in the first expression with each term in the second expression.

(−2d2+s)(5d2−6s)=(−2d2)(5d2)+(−2d2)(−6s)+(s)(5d2)+(s)(−6s)(-2d^2 + s)(5d^2 - 6s) = (-2d^2)(5d^2) + (-2d^2)(-6s) + (s)(5d^2) + (s)(-6s)

Step 2: Simplify the Expression

Now, we will simplify the expression by multiplying the terms.

(−2d2)(5d2)=−10d4(-2d^2)(5d^2) = -10d^4

(−2d2)(−6s)=12d2s(-2d^2)(-6s) = 12d^2s

(s)(5d2)=5d2s(s)(5d^2) = 5d^2s

(s)(−6s)=−6s2(s)(-6s) = -6s^2

Step 3: Combine Like Terms

We will now combine the like terms to simplify the expression further.

−10d4+12d2s+5d2s−6s2-10d^4 + 12d^2s + 5d^2s - 6s^2

Step 4: Final Simplification

Combining the like terms, we get:

−10d4+17d2s−6s2-10d^4 + 17d^2s - 6s^2

Conclusion

The product of the given expression is −10d4+17d2s−6s2-10d^4 + 17d^2s - 6s^2. This matches option A.

Discussion

The given problem involves finding the product of two quadratic expressions. We applied the distributive property to multiply the expressions and then simplified the resulting expression by combining like terms. This problem requires a good understanding of algebraic expressions and the distributive property.

Key Takeaways

  • The distributive property states that for any real numbers a, b, and c, a(b + c) = ab + ac.
  • To find the product of two quadratic expressions, we can apply the distributive property by multiplying each term in the first expression with each term in the second expression.
  • We can simplify the resulting expression by combining like terms.

Final Answer

The final answer is −10d4+17d2s−6s2\boxed{-10d^4 + 17d^2s - 6s^2}.

Understanding Quadratic Expressions

Quadratic expressions are a type of polynomial expression that contains a squared variable. They are commonly used in algebra and are an essential part of mathematics. In this article, we will discuss the product of quadratic expressions and answer some frequently asked questions.

Q&A

Q1: What is the product of two quadratic expressions?

A1: The product of two quadratic expressions is a new quadratic expression that is obtained by multiplying the two given expressions. To find the product, we need to apply the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac.

Q2: How do I apply the distributive property to find the product of two quadratic expressions?

A2: To apply the distributive property, we need to multiply each term in the first expression with each term in the second expression. This will result in a new expression that contains multiple terms.

Q3: What is the difference between the product of two quadratic expressions and the sum of two quadratic expressions?

A3: The product of two quadratic expressions is obtained by multiplying the two expressions, while the sum of two quadratic expressions is obtained by adding the two expressions. For example, if we have two quadratic expressions x^2 + 2x and x^2 - 3x, the product would be (x^2 + 2x)(x^2 - 3x), while the sum would be (x^2 + 2x) + (x^2 - 3x).

Q4: Can I use the distributive property to find the product of more than two quadratic expressions?

A4: Yes, you can use the distributive property to find the product of more than two quadratic expressions. However, it can become complex and time-consuming. In such cases, it is recommended to use the FOIL method, which is a shortcut for finding the product of two binomials.

Q5: What is the FOIL method?

A5: The FOIL method is a shortcut for finding the product of two binomials. It stands for First, Outer, Inner, Last, and it involves multiplying the first terms, then the outer terms, then the inner terms, and finally the last terms.

Q6: Can I use the distributive property to find the product of a quadratic expression and a linear expression?

A6: Yes, you can use the distributive property to find the product of a quadratic expression and a linear expression. This will result in a new expression that contains a quadratic term and a linear term.

Q7: What is the difference between the product of a quadratic expression and a linear expression and the product of two quadratic expressions?

A7: The product of a quadratic expression and a linear expression is obtained by multiplying the quadratic expression with the linear expression, while the product of two quadratic expressions is obtained by multiplying the two quadratic expressions. For example, if we have a quadratic expression x^2 + 2x and a linear expression x + 1, the product would be (x^2 + 2x)(x + 1), while the product of two quadratic expressions would be (x^2 + 2x)(x^2 - 3x).

Q8: Can I use the distributive property to find the product of a quadratic expression and a constant?

A8: Yes, you can use the distributive property to find the product of a quadratic expression and a constant. This will result in a new expression that contains a quadratic term.

Q9: What is the difference between the product of a quadratic expression and a constant and the product of two quadratic expressions?

A9: The product of a quadratic expression and a constant is obtained by multiplying the quadratic expression with the constant, while the product of two quadratic expressions is obtained by multiplying the two quadratic expressions. For example, if we have a quadratic expression x^2 + 2x and a constant 3, the product would be 3(x^2 + 2x), while the product of two quadratic expressions would be (x^2 + 2x)(x^2 - 3x).

Q10: Can I use the distributive property to find the product of a quadratic expression and a polynomial expression of higher degree?

A10: Yes, you can use the distributive property to find the product of a quadratic expression and a polynomial expression of higher degree. However, it can become complex and time-consuming. In such cases, it is recommended to use the FOIL method or to simplify the expression by combining like terms.

Conclusion

In this article, we have discussed the product of quadratic expressions and answered some frequently asked questions. We have also provided examples and explanations to help you understand the concept better. We hope that this article has been helpful in clarifying any doubts you may have had about the product of quadratic expressions.