If [tex]$m(x)=x^2+3$[/tex] and [tex]$n(x)=5x+9$[/tex], which expression is equivalent to [tex]$(m n)(x)$[/tex]?
A. [tex]$5 x^3+9 x^2+15 x+27$[/tex]
B. [tex]$25 x^2+90 x+84$[/tex]
C. [tex]$x^2+5 x+12$[/tex]
D. [tex]$5 x^2+24$[/tex]
Answer (2)
The expression equivalent to ( mn ) ( x ) is obtained by multiplying the two functions, resulting in 5 x 3 + 9 x 2 + 15 x + 27 , which matches option A.
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Multiply the two functions: ( mn ) ( x ) = ( x 2 + 3 ) ( 5 x + 9 ) .
Expand the product: ( x 2 + 3 ) ( 5 x + 9 ) = x 2 ( 5 x + 9 ) + 3 ( 5 x + 9 ) .
Distribute: x 2 ( 5 x + 9 ) + 3 ( 5 x + 9 ) = 5 x 3 + 9 x 2 + 15 x + 27 .
The final expression is: 5 x 3 + 9 x 2 + 15 x + 27 .
Explanation
Understanding the Problem We are given two functions, m ( x ) = x 2 + 3 and n ( x ) = 5 x + 9 . We need to find the expression that is equivalent to the product of these two functions, which is denoted as ( mn ) ( x ) . This means we need to multiply m ( x ) and n ( x ) together.
Multiplying the Functions To find ( mn ) ( x ) , we multiply the two functions: ( mn ) ( x ) = m ( x ) ( x ) = ( x 2 + 3 ) ( 5 x + 9 ) Now, we expand the product by distributing each term in the first parenthesis to each term in the second parenthesis:
Expanding and Distributing Expanding the product, we get: ( x 2 + 3 ) ( 5 x + 9 ) = x 2 ( 5 x + 9 ) + 3 ( 5 x + 9 ) Now, distribute x 2 and 3 to the terms inside their respective parentheses: x 2 ( 5 x + 9 ) = 5 x 3 + 9 x 2 3 ( 5 x + 9 ) = 15 x + 27 So, we have: ( mn ) ( x ) = 5 x 3 + 9 x 2 + 15 x + 27
Final Expression The resulting expression after multiplying and simplifying is 5 x 3 + 9 x 2 + 15 x + 27 .
Examples
Understanding function multiplication is useful in many areas. For example, if you know the area of a rectangular garden is given by x 2 + 3 and the number of plants you can fit per unit area is 5 x + 9 , then the total number of plants you can grow is given by the product of these two functions. This helps in planning and resource allocation.
