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 ($l w$) 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 (1)
Multiply the expressions for length and width: ( 3 x 2 + 5 x + 10 ) ( x 2 − 3 x − 1 ) .
Expand the expression: 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 the patio is represented by the expression: 3 x 4 − 4 x 3 − 8 x 2 − 35 x − 10 .
Explanation
Understanding the Problem Vanessa represents the length of her patio as ( 3 x 2 + 5 x + 10 ) and the width as ( x 2 − 3 x − 1 ) . To find the area of the patio, we need to multiply these two expressions.
Expanding the Expression We need to multiply the two expressions: ( 3 x 2 + 5 x + 10 ) ( x 2 − 3 x − 1 ) . Let's expand this product:
3 x 2 ( x 2 − 3 x − 1 ) + 5 x ( x 2 − 3 x − 1 ) + 10 ( x 2 − 3 x − 1 )
Distributing Now, let's distribute each term:
3 x 2 ( x 2 ) + 3 x 2 ( − 3 x ) + 3 x 2 ( − 1 ) + 5 x ( x 2 ) + 5 x ( − 3 x ) + 5 x ( − 1 ) + 10 ( x 2 ) + 10 ( − 3 x ) + 10 ( − 1 )
= 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, let's 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 The expression representing the area of Vanessa's patio is 3 x 4 − 4 x 3 − 8 x 2 − 35 x − 10 .
Examples
Understanding polynomial multiplication is crucial in various fields, such as engineering and computer graphics. For instance, when designing a rectangular garden where the length is represented by ( 3 x 2 + 5 x + 10 ) and the width by ( x 2 − 3 x − 1 ) , determining the area involves multiplying these expressions. This calculation helps in estimating the amount of soil needed, planning the layout, and optimizing the use of space. Similarly, in computer graphics, polynomial expressions are used to define curves and surfaces, and multiplying these expressions helps in rendering complex shapes and scenes.
