The domain for [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex] is the set of all real numbers.

Let [tex]$f(x)=2 x^2+x-3$[/tex] and [tex]$g(x)=x+2$[/tex]
Find [tex]$(f \cdot g)(x)$[/tex].

Multiply the two functions: ( f \tcdot g ) ( x ) = ( 2 x 2 + x − 3 ) ( x + 2 ) .
Expand the product: 2 x 3 + 4 x 2 + x 2 + 2 x − 3 x − 6 .
Combine like terms: 2 x 3 + 5 x 2 − x − 6 .
The product of the functions is 2 x 3 + 5 x 2 − x − 6 ​ .

Explanation

Understanding the Problem We are given two functions, f ( x ) = 2 x 2 + x − 3 and g ( x ) = x + 2 , and we need to find their product, ( f \tcdot g ) ( x ) . This means we need to multiply the two expressions together.

Multiplying the Functions To find the product of the two functions, we multiply them: ( f \tcdot g ) ( x ) = f ( x ) \tcdot g ( x ) = ( 2 x 2 + x − 3 ) ( x + 2 ) Now, we expand the product by multiplying each term in the first expression by each term in the second expression:

Expanding the Product Expanding the product: ( 2 x 2 + x − 3 ) ( x + 2 ) = 2 x 2 ( x ) + 2 x 2 ( 2 ) + x ( x ) + x ( 2 ) − 3 ( x ) − 3 ( 2 ) = 2 x 3 + 4 x 2 + x 2 + 2 x − 3 x − 6

Simplifying the Expression Now, we combine like terms to simplify the expression: 2 x 3 + 4 x 2 + x 2 + 2 x − 3 x − 6 = 2 x 3 + ( 4 x 2 + x 2 ) + ( 2 x − 3 x ) − 6 = 2 x 3 + 5 x 2 − x − 6

Final Answer Therefore, the product of the two functions is: ( f \tcdot g ) ( x ) = 2 x 3 + 5 x 2 − x − 6 So, the correct answer is 2 x 3 + 5 x 2 − x − 6 .


Examples
Understanding polynomial multiplication is crucial in various fields, such as engineering, computer science, and economics. For instance, when designing a bridge, engineers use polynomial functions to model the load distribution and stress on different parts of the structure. Multiplying these polynomials helps them predict the overall behavior of the bridge under various conditions. Similarly, in economics, polynomial functions can represent cost and revenue models, and their multiplication can help analyze profit margins and optimize business strategies. In computer graphics, polynomial multiplication is used to create complex curves and surfaces for realistic rendering.

Answered by GinnyAnswer | 2025-07-03

The product of the functions f ( x ) = 2 x 2 + x − 3 and g ( x ) = x + 2 is 2 x 3 + 5 x 2 − x − 6 .
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Answered by Anonymous | 2025-07-04