Multiply $(x^2+3 x+4)(3 x^2-2 x+1)$.
A. $3 x^4+11 x^3+19 x^2+11 x+4$
B. $4 x^2+x+5$
C. $3 x^4+7 x^3+7 x^2-5 x+4$
D. $3 x^4-6 x^2+4$
Answer (2)
Distribute each term of the first polynomial to each term of the second polynomial.
Expand the product: ( x 2 + 3 x + 4 ) ( 3 x 2 − 2 x + 1 ) = 3 x 4 − 2 x 3 + x 2 + 9 x 3 − 6 x 2 + 3 x + 12 x 2 − 8 x + 4 .
Combine like terms: 3 x 4 + 7 x 3 + 7 x 2 − 5 x + 4 .
The final result is 3 x 4 + 7 x 3 + 7 x 2 − 5 x + 4 .
Explanation
Understanding the Problem We are asked to multiply two polynomials: ( x 2 + 3 x + 4 ) ( 3 x 2 − 2 x + 1 ) . This involves distributing each term of the first polynomial across each term of the second polynomial and then combining like terms.
Expanding the Product Let's multiply each term of the first polynomial by each term of the second polynomial:
( x 2 + 3 x + 4 ) ( 3 x 2 − 2 x + 1 ) = x 2 ( 3 x 2 − 2 x + 1 ) + 3 x ( 3 x 2 − 2 x + 1 ) + 4 ( 3 x 2 − 2 x + 1 )
Distributing Now, distribute each term:
x 2 ( 3 x 2 − 2 x + 1 ) = 3 x 4 − 2 x 3 + x 2
3 x ( 3 x 2 − 2 x + 1 ) = 9 x 3 − 6 x 2 + 3 x
4 ( 3 x 2 − 2 x + 1 ) = 12 x 2 − 8 x + 4
Combining Terms Next, we add the resulting terms together:
( 3 x 4 − 2 x 3 + x 2 ) + ( 9 x 3 − 6 x 2 + 3 x ) + ( 12 x 2 − 8 x + 4 )
Simplifying Now, combine like terms:
3 x 4 + ( − 2 x 3 + 9 x 3 ) + ( x 2 − 6 x 2 + 12 x 2 ) + ( 3 x − 8 x ) + 4
3 x 4 + 7 x 3 + 7 x 2 − 5 x + 4
Final Result So, the product of the two polynomials is 3 x 4 + 7 x 3 + 7 x 2 − 5 x + 4 .
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 rational function, which involves polynomial multiplication. Multiplying polynomials helps in analyzing the stability and performance of the system. Also, in computer graphics, polynomial multiplication is used in curve and surface design.
To multiply the polynomials ( x 2 + 3 x + 4 ) ( 3 x 2 − 2 x + 1 ) , we distribute each term and combine like terms to get 3 x 4 + 7 x 3 + 7 x 2 − 5 x + 4 . The correct answer is option C. This method illustrates the process of polynomial multiplication clearly and step-by-step.
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