Multiply the polynomials.
$\left(2 x^2+6 x+6\right)(3 x-2)$
A. $6 x^3+14 x^2+6 x-12$
B. $6 x^3-14 x^2+6 x+12$
C. $6 x^3+14 x^2+6 x+12$
D. $6 x^3-22 x^2-30 x-12$
Answer (1)
To multiply the polynomials ( 2 x 2 + 6 x + 6 ) and ( 3 x − 2 ) :
Multiply 2 x 2 by ( 3 x − 2 ) to get 6 x 3 − 4 x 2 .
Multiply 6 x by ( 3 x − 2 ) to get 18 x 2 − 12 x .
Multiply 6 by ( 3 x − 2 ) to get 18 x − 12 .
Add the results and combine like terms: 6 x 3 + 14 x 2 + 6 x − 12 .
The final answer is 6 x 3 + 14 x 2 + 6 x − 12 .
Explanation
Problem Analysis We need to multiply the polynomials ( 2 x 2 + 6 x + 6 ) and ( 3 x − 2 ) . We will use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.
Multiplying the first term First, we multiply 2 x 2 by ( 3 x − 2 ) : 2 x 2 ( 3 x − 2 ) = 2 x 2 ( 3 x ) + 2 x 2 ( − 2 ) = 6 x 3 − 4 x 2
Multiplying the second term Next, we multiply 6 x by ( 3 x − 2 ) : 6 x ( 3 x − 2 ) = 6 x ( 3 x ) + 6 x ( − 2 ) = 18 x 2 − 12 x
Multiplying the third term Then, we multiply 6 by ( 3 x − 2 ) : 6 ( 3 x − 2 ) = 6 ( 3 x ) + 6 ( − 2 ) = 18 x − 12
Adding the results Now, we add the results together: ( 6 x 3 − 4 x 2 ) + ( 18 x 2 − 12 x ) + ( 18 x − 12 ) = 6 x 3 − 4 x 2 + 18 x 2 − 12 x + 18 x − 12
Combining like terms Finally, we combine like terms: 6 x 3 + ( − 4 x 2 + 18 x 2 ) + ( − 12 x + 18 x ) − 12 = 6 x 3 + 14 x 2 + 6 x − 12
Final Answer The result of the multiplication is 6 x 3 + 14 x 2 + 6 x − 12 . Comparing this with the given options, we see that it matches option A.
Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, in control systems, the transfer function of a system can be represented as a ratio of two polynomials. Multiplying these polynomials helps in analyzing the system's behavior and designing controllers. In computer graphics, polynomial multiplication is used in curve and surface modeling to create smooth and realistic shapes.
