Simplify the following: $\left(2 x^2+3 x-3\right)\left(3 x^2+2 x+3\right)$
Answer (1)
Expand the product of the two polynomials: ( 2 x 2 + 3 x − 3 ) ( 3 x 2 + 2 x + 3 ) .
Distribute and multiply each term.
Combine like terms to simplify the expression.
The simplified expression is 6 x 4 + 13 x 3 + 3 x 2 + 3 x − 9 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 2 x 2 + 3 x − 3 ) ( 3 x 2 + 2 x + 3 ) . This involves expanding the product of the two polynomials and combining like terms.
Expanding the Product To expand the product, we multiply each term in the first polynomial by each term in the second polynomial:
( 2 x 2 + 3 x − 3 ) ( 3 x 2 + 2 x + 3 ) = 2 x 2 ( 3 x 2 + 2 x + 3 ) + 3 x ( 3 x 2 + 2 x + 3 ) − 3 ( 3 x 2 + 2 x + 3 )
Distributing the Terms Now, we distribute each term:
2 x 2 ( 3 x 2 + 2 x + 3 ) = 6 x 4 + 4 x 3 + 6 x 2
3 x ( 3 x 2 + 2 x + 3 ) = 9 x 3 + 6 x 2 + 9 x
− 3 ( 3 x 2 + 2 x + 3 ) = − 9 x 2 − 6 x − 9
Combining Like Terms Next, we add the results together:
( 6 x 4 + 4 x 3 + 6 x 2 ) + ( 9 x 3 + 6 x 2 + 9 x ) + ( − 9 x 2 − 6 x − 9 ) = 6 x 4 + ( 4 x 3 + 9 x 3 ) + ( 6 x 2 + 6 x 2 − 9 x 2 ) + ( 9 x − 6 x ) − 9
Combining like terms, we get:
6 x 4 + 13 x 3 + 3 x 2 + 3 x − 9
Final Answer The simplified expression is 6 x 4 + 13 x 3 + 3 x 2 + 3 x − 9 . The question asks us to express this in the form □ x . However, we cannot factor out x from all the terms in the polynomial since the constant term is -9. Therefore, we cannot write the expression in the required form.
Conclusion Since we cannot express the polynomial in the form □ x , there must have been a misunderstanding. The simplified form of the given expression is:
6 x 4 + 13 x 3 + 3 x 2 + 3 x − 9
Examples
Polynomial multiplication is a fundamental concept in algebra and is used in various fields such as engineering, physics, and computer science. For example, when designing a bridge, engineers use polynomial functions to model the load distribution and stress on different parts of the structure. Multiplying these polynomials helps them understand the combined effect of different factors and ensure the bridge's stability. Similarly, in computer graphics, polynomial multiplication is used to create complex shapes and textures by combining simpler geometric forms.
