Vanessa uses the expressions $\left(3 x^2+5 x+10\right)$ and $\left(x^2-3 x-1\right)$ to represent the length and width of her patio. Which expression represents the area of Vanessa's patio?
A. $3 x^4-4 x^3-8 x^2-35 x-10$
B. $3 x^4+14 x^3+22 x^2+25 x-10$
C. $3 x^4-4 x^3+22 x^2+35 x+10$
D. $3 x^4+14 x^3+28 x^2+35 x+10$
Answer (2)
Multiply the expressions for length and width: ( 3 x 2 + 5 x + 10 ) ( x 2 − 3 x − 1 ) .
Expand the product: 3 x 4 − 9 x 3 − 3 x 2 + 5 x 3 − 15 x 2 − 5 x + 10 x 2 − 30 x − 10 .
Combine like terms: 3 x 4 − 4 x 3 − 8 x 2 − 35 x − 10 .
The area of Vanessa's patio is 3 x 4 − 4 x 3 − 8 x 2 − 35 x − 10 .
Explanation
Understanding the Problem The problem states that Vanessa uses the expressions ( 3 x 2 + 5 x + 10 ) and ( x 2 − 3 x − 1 ) to represent the length and width of her patio. We need to find the expression that represents the area of the patio, which is the product of the length and width.
Setting up the Multiplication To find the area, we need to multiply the two expressions: ( 3 x 2 + 5 x + 10 ) ( x 2 − 3 x − 1 )
Expanding the Product Now, let's expand the product by multiplying each term in the first expression by each term in the second expression:
3 x 2 ( x 2 − 3 x − 1 ) + 5 x ( x 2 − 3 x − 1 ) + 10 ( x 2 − 3 x − 1 )
Expanding each term:
3 x 4 − 9 x 3 − 3 x 2 + 5 x 3 − 15 x 2 − 5 x + 10 x 2 − 30 x − 10
Combining Like Terms Now, we combine like terms:
3 x 4 + ( − 9 x 3 + 5 x 3 ) + ( − 3 x 2 − 15 x 2 + 10 x 2 ) + ( − 5 x − 30 x ) − 10
3 x 4 − 4 x 3 − 8 x 2 − 35 x − 10
Final Answer Therefore, the expression that represents the area of Vanessa's patio is:
3 x 4 − 4 x 3 − 8 x 2 − 35 x − 10
Examples
Understanding polynomial multiplication is essential in various fields, such as engineering and computer graphics. For example, when designing a rectangular garden, you might use polynomial expressions to represent the length and width. Multiplying these expressions gives you the area, helping you determine the amount of soil needed or the size of the fence required. This concept extends to more complex shapes and volumes, making it a fundamental tool in spatial planning and resource management.
The area of Vanessa's patio is given by the expression 3 x 4 − 4 x 3 − 8 x 2 − 35 x − 10 , which results from multiplying the length and width expressions. This corresponds to option A.
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