Select the correct answer.
Consider functions [tex]$f$[/tex] and [tex]$g$[/tex].
[tex]$\begin{array}{l}
f(x)=4 x^2+3 x \\
g(x)=-4 x-6
\end{array}$[/tex]
Which expression is equal to [tex]$f(x) \cdot g(x)$[/tex]?
A. [tex]$-16 x^3-36 x^2-18 x$[/tex]
B. [tex]$-16 x^3+18 x$[/tex]
C. [tex]$-16 x^3-36 x^2+18 x$[/tex]
D. [tex]$-16 x^3+12 x^2-18 x$[/tex]
Answer (2)
Multiply the two polynomials f ( x ) and g ( x ) : ( 4 x 2 + 3 x ) ( − 4 x − 6 ) .
Distribute and expand the expression: 4 x 2 ( − 4 x − 6 ) + 3 x ( − 4 x − 6 ) = − 16 x 3 − 24 x 2 − 12 x 2 − 18 x .
Combine like terms: − 16 x 3 − 36 x 2 − 18 x .
The expression equal to f ( x ) \tims g ( x ) is − 16 x 3 − 36 x 2 − 18 x .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 4 x 2 + 3 x and g ( x ) = − 4 x − 6 . Our goal is to find the expression that is equal to the product of these two functions, f ( x ) \tims g ( x ) .
Setting up the Multiplication To find the product f ( x ) \tims g ( x ) , we need to multiply the two expressions together: ( 4 x 2 + 3 x ) ( − 4 x − 6 ) We will use the distributive property to expand this expression.
Expanding the Expression First, distribute 4 x 2 over ( − 4 x − 6 ) : 4 x 2 ( − 4 x − 6 ) = 4 x 2 ( − 4 x ) + 4 x 2 ( − 6 ) = − 16 x 3 − 24 x 2 Next, distribute 3 x over ( − 4 x − 6 ) : 3 x ( − 4 x − 6 ) = 3 x ( − 4 x ) + 3 x ( − 6 ) = − 12 x 2 − 18 x Now, combine these two results:
Combining Like Terms Adding the two expressions we obtained: ( − 16 x 3 − 24 x 2 ) + ( − 12 x 2 − 18 x ) = − 16 x 3 − 24 x 2 − 12 x 2 − 18 x Combine the like terms (the x 2 terms): − 24 x 2 − 12 x 2 = − 36 x 2 So the final expression is: − 16 x 3 − 36 x 2 − 18 x
Final Answer Comparing our result with the given options, we see that the correct expression is − 16 x 3 − 36 x 2 − 18 x .
Examples
Understanding polynomial multiplication is crucial in various fields, such as physics and engineering, where complex systems are modeled using polynomial equations. For instance, when analyzing the trajectory of a projectile, multiplying polynomial functions helps determine its path and landing point. Similarly, in electrical engineering, polynomial multiplication is used to analyze circuits and design filters. This mathematical operation allows engineers and scientists to make accurate predictions and optimize system performance.
We multiplied the given functions f ( x ) = 4 x 2 + 3 x and g ( x ) = − 4 x − 6 using the distributive property. The resulting expression is − 16 x 3 − 36 x 2 − 18 x , which corresponds to option A. Therefore, the correct answer is − 16 x 3 − 36 x 2 − 18 x .
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