What is the product of [tex]$\left(2 x^2+5 x-3\right)$[/tex] and [tex]$(3 x+7)$[/tex]?
A. [tex]$6 x^3+29 x^2-26 x+21$[/tex]
B. [tex]$6 x^3+14 x^2+35 x-21$[/tex]
C. [tex]$6 x^3+15 x^2-9 x-21$[/tex]
D. [tex]$6 x^3+29 x^2+26 x-21$[/tex]
Answer (1)
Distribute ( 3 x + 7 ) over ( 2 x 2 + 5 x − 3 ) .
Multiply each term: 2 x 2 ( 3 x + 7 ) + 5 x ( 3 x + 7 ) − 3 ( 3 x + 7 ) .
Expand and combine like terms: 6 x 3 + 14 x 2 + 15 x 2 + 35 x − 9 x − 21 .
Simplify to get the final product: 6 x 3 + 29 x 2 + 26 x − 21 .
Explanation
Understanding the Problem We need to find the product of the two polynomials ( 2 x 2 + 5 x − 3 ) and ( 3 x + 7 ) . This involves multiplying each term of the first polynomial by each term of the second polynomial and then combining like terms.
Applying the Distributive Property We will use the distributive property to multiply the two polynomials:
( 2 x 2 + 5 x − 3 ) ( 3 x + 7 ) = 2 x 2 ( 3 x + 7 ) + 5 x ( 3 x + 7 ) − 3 ( 3 x + 7 )
Expanding Each Term Now, we expand each term:
2 x 2 ( 3 x + 7 ) = 6 x 3 + 14 x 2
5 x ( 3 x + 7 ) = 15 x 2 + 35 x
− 3 ( 3 x + 7 ) = − 9 x − 21
Combining the Terms Next, we combine the expanded terms:
6 x 3 + 14 x 2 + 15 x 2 + 35 x − 9 x − 21
Simplifying the Expression Finally, we simplify by combining like terms:
6 x 3 + ( 14 x 2 + 15 x 2 ) + ( 35 x − 9 x ) − 21 = 6 x 3 + 29 x 2 + 26 x − 21
Identifying the Correct Answer Comparing our result with the given options, we find that the correct answer is:
6 x 3 + 29 x 2 + 26 x − 21
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 essential for solving real-world problems in these fields.
