Multiply the polynomials.
$(x+2)\left(2 x^2+9 x+8\right)$
A. $16 x^3+72 x^2+46 x-16$
B. $2 x^3+13 x^2-26 x+16$
C. $2 x^3+17 x^2+22 x+16$
D. $2 x^3+13 x^2+26 x+16$
Answer (1)
Distribute each term of ( x + 2 ) to ( 2 x 2 + 9 x + 8 ) .
Expand the terms: x ( 2 x 2 + 9 x + 8 ) + 2 ( 2 x 2 + 9 x + 8 ) = 2 x 3 + 9 x 2 + 8 x + 4 x 2 + 18 x + 16 .
Combine like terms: 2 x 3 + ( 9 x 2 + 4 x 2 ) + ( 8 x + 18 x ) + 16 = 2 x 3 + 13 x 2 + 26 x + 16 .
The correct answer is 2 x 3 + 13 x 2 + 26 x + 16 , which corresponds to option D. D
Explanation
Understanding the Problem We are given two polynomials, ( x + 2 ) and ( 2 x 2 + 9 x + 8 ) , and we need to find their product. This involves multiplying each term of the first polynomial by each term of the second polynomial and then combining like terms.
Distributing the Terms To multiply the polynomials ( x + 2 ) and ( 2 x 2 + 9 x + 8 ) , we distribute each term of ( x + 2 ) to each term of ( 2 x 2 + 9 x + 8 ) . This gives us:
x ( 2 x 2 + 9 x + 8 ) + 2 ( 2 x 2 + 9 x + 8 )
Expanding the Terms Now, we expand each term:
x ( 2 x 2 ) + x ( 9 x ) + x ( 8 ) + 2 ( 2 x 2 ) + 2 ( 9 x ) + 2 ( 8 )
= 2 x 3 + 9 x 2 + 8 x + 4 x 2 + 18 x + 16
Combining Like Terms Next, we combine like terms:
2 x 3 + ( 9 x 2 + 4 x 2 ) + ( 8 x + 18 x ) + 16
= 2 x 3 + 13 x 2 + 26 x + 16
Selecting the Correct Option Finally, we compare our result, 2 x 3 + 13 x 2 + 26 x + 16 , with the given options. Option D matches our result exactly.
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. Understanding polynomial multiplication is fundamental for solving many real-world problems in these areas.
