Dylan uses the expressions [tex]$\left(x^2-2 x+8\right)$[/tex] and [tex]$\left(2 x^2+5 x-7\right)$[/tex] to represent the length and width of his bedroom. Which expression represents the area (lw) of Dylan's room?
A. [tex]$2 x^4-10 x^2-56$[/tex]
B. [tex]$2 x^4+10 x^2+56$[/tex]
C. [tex]$2 x^4+x^3-x^2+54 x-56$[/tex]
D. [tex]$2 x^4+9 x^3+33 x^2+54 x+56$[/tex]
Answer (2)
Multiply the expressions for length and width: ( x 2 − 2 x + 8 ) ( 2 x 2 + 5 x − 7 ) .
Expand the expression: 2 x 4 + 5 x 3 − 7 x 2 − 4 x 3 − 10 x 2 + 14 x + 16 x 2 + 40 x − 56 .
Combine like terms: 2 x 4 + x 3 − x 2 + 54 x − 56 .
The area of Dylan's room is 2 x 4 + x 3 − x 2 + 54 x − 56 .
Explanation
Understanding the Problem We are given the length and width of Dylan's room as expressions in terms of x . The length is ( x 2 − 2 x + 8 ) and the width is ( 2 x 2 + 5 x − 7 ) . To find the area of the room, we need to multiply these two expressions.
Setting up the Area Calculation The area of Dylan's room is given by the product of the length and width: A re a = ( x 2 − 2 x + 8 ) ( 2 x 2 + 5 x − 7 ) We need to expand this expression by multiplying each term in the first expression by each term in the second expression.
Expanding the Expression Let's expand the expression:
A re a = x 2 ( 2 x 2 + 5 x − 7 ) − 2 x ( 2 x 2 + 5 x − 7 ) + 8 ( 2 x 2 + 5 x − 7 ) = ( 2 x 4 + 5 x 3 − 7 x 2 ) − ( 4 x 3 + 10 x 2 − 14 x ) + ( 16 x 2 + 40 x − 56 ) = 2 x 4 + 5 x 3 − 7 x 2 − 4 x 3 − 10 x 2 + 14 x + 16 x 2 + 40 x − 56
Combining Like Terms Now, we combine like terms:
A re a = 2 x 4 + ( 5 x 3 − 4 x 3 ) + ( − 7 x 2 − 10 x 2 + 16 x 2 ) + ( 14 x + 40 x ) − 56 = 2 x 4 + x 3 + ( − 17 x 2 + 16 x 2 ) + 54 x − 56 = 2 x 4 + x 3 − x 2 + 54 x − 56
Final Answer Therefore, the expression representing the area of Dylan's room is 2 x 4 + x 3 − x 2 + 54 x − 56 .
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 and width are expressed as polynomials, calculating the area involves multiplying these polynomials. If the length is ( x 2 − 2 x + 8 ) meters and the width is ( 2 x 2 + 5 x − 7 ) meters, the area, found to be 2 x 4 + x 3 − x 2 + 54 x − 56 square meters, helps determine the amount of fencing needed or the quantity of soil required. This concept extends to more complex scenarios, like optimizing resource allocation in urban planning or modeling physical spaces in architectural design.
The area of Dylan's room can be calculated by multiplying the length x 2 − 2 x + 8 by the width 2 x 2 + 5 x − 7 . The resulting expression for the area is 2 x 4 + x 3 − x 2 + 54 x − 56 . Thus, the option chosen is C.
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