$\left(3 c^2+2 d\right)\left(-5 c^2+d\right)$
Select all of the partial products for the multiplication problem above.
$2 d^2$
$3 c d^3$
$-15 c^4$
$-10 c^2 d$
$-15 c ^2$
$3 c^2 d$
Answer (2)
Expand the expression ( 3 c 2 + 2 d ) ( − 5 c 2 + d ) using the distributive property.
Identify the individual terms before simplification as partial products: − 15 c 4 , 3 c 2 d , − 10 c 2 d , and 2 d 2 .
Compare the calculated partial products with the given options.
Select the correct partial products: 2 d 2 , − 15 c 4 , − 10 c 2 d , and 3 c 2 d .
Explanation
Understanding the Problem We are given the expression ( 3 c 2 + 2 d ) ( − 5 c 2 + d ) and asked to identify the partial products of the multiplication.
Expanding the Expression To find the partial products, we need to expand the expression using the distributive property (also known as the FOIL method). We multiply each term in the first parenthesis by each term in the second parenthesis.
Calculating Partial Products First, multiply 3 c 2 by both terms in the second parenthesis:
3 c 2 ∗ ( − 5 c 2 ) = − 15 c 4 3 c 2 ∗ ( d ) = 3 c 2 d
Next, multiply 2 d by both terms in the second parenthesis:
2 d ∗ ( − 5 c 2 ) = − 10 c 2 d 2 d ∗ ( d ) = 2 d 2
Identifying Partial Products So the expanded expression is: − 15 c 4 + 3 c 2 d − 10 c 2 d + 2 d 2 . The partial products are the individual terms before simplification.
Selecting Correct Options Now, we compare the partial products we found with the given options:
2 d 2 - This is a partial product. 3 c d 3 - This is NOT a partial product. − 15 c 4 - This is a partial product. − 10 c 2 d - This is a partial product. − 15 c 2 - This is NOT a partial product. 3 c 2 d - This is a partial product.
Final Answer Therefore, the correct partial products are 2 d 2 , − 15 c 4 , − 10 c 2 d , and 3 c 2 d .
Examples
Understanding partial products is essential when calculating areas or volumes in geometry. For example, if you're designing a rectangular garden with dimensions ( 3 x + 2 ) and ( 4 x + 1 ) , the partial products from multiplying these expressions represent the individual areas of smaller sections within the garden. Calculating these partial areas helps in determining the total area, which is crucial for purchasing the correct amount of soil or fencing. This concept extends to more complex shapes and volumes, making it a fundamental skill in design and resource management.
The partial products from multiplying ( 3 c 2 + 2 d ) ( − 5 c 2 + d ) are 2 d 2 , − 15 c 4 , − 10 c 2 d , and 3 c 2 d .
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