Using the distributive property, Marta multiplied the binomial $(2 x+3)$ by the trinomial $\left(x^2+x-2\right)$ and got the expression below.
$(2 x)\left(x^2\right)+(2 x)(x)+(2 x)(-2)+(3)\left(x^2\right)+(3)(x)+(3)(-2)$
Which is the simplified product?
A. $2 x^3+6 x^2-x-6$
B. $2 x^3+x^2-x-6$
C. $2 x^3+5 x^2-x-6$
D. $2 x^3-x^2-7 x-6$
Answer (1)
Expand each term in the expression: ( 2 x ) ( x 2 ) + ( 2 x ) ( x ) + ( 2 x ) ( − 2 ) + ( 3 ) ( x 2 ) + ( 3 ) ( x ) + ( 3 ) ( − 2 ) .
Simplify each term: 2 x 3 + 2 x 2 − 4 x + 3 x 2 + 3 x − 6 .
Combine like terms: 2 x 3 + ( 2 x 2 + 3 x 2 ) + ( − 4 x + 3 x ) − 6 .
The simplified product is 2 x 3 + 5 x 2 − x − 6 .
Explanation
Understanding the Expression We are given the expression ( 2 x ) ( x 2 ) + ( 2 x ) ( x ) + ( 2 x ) ( − 2 ) + ( 3 ) ( x 2 ) + ( 3 ) ( x ) + ( 3 ) ( − 2 ) and we want to simplify it. This involves expanding each term and then combining like terms.
Expanding Each Term First, let's expand each term:
( 2 x ) ( x 2 ) = 2 x 3 ( 2 x ) ( x ) = 2 x 2 ( 2 x ) ( − 2 ) = − 4 x ( 3 ) ( x 2 ) = 3 x 2 ( 3 ) ( x ) = 3 x ( 3 ) ( − 2 ) = − 6
So the expression becomes: 2 x 3 + 2 x 2 − 4 x + 3 x 2 + 3 x − 6
Combining Like Terms Now, let's combine like terms. We have:
2 x 3 (only one term with x 3 )
2 x 2 + 3 x 2 = 5 x 2 − 4 x + 3 x = − x − 6 (constant term)
So, the simplified expression is: 2 x 3 + 5 x 2 − x − 6
Final Answer Comparing our simplified expression 2 x 3 + 5 x 2 − x − 6 with the given options, we see that it matches the third option.
Therefore, the simplified product is 2 x 3 + 5 x 2 − x − 6 .
Examples
Polynomial multiplication is used in various fields, such as engineering, physics, and computer graphics. 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.
