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.
Combine like terms.
The simplified product is 2 x 3 + 5 x 2 − x − 6 .
Explanation
Understanding the Problem We are given the expression ( 2 x ) ( x 2 ) + ( 2 x ) ( x ) + ( 2 x ) ( − 2 ) + ( 3 ) ( x 2 ) + ( 3 ) ( x ) + ( 3 ) ( − 2 ) and asked 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
Rewriting the Expression Now, let's rewrite the expression with the expanded terms:
2 x 3 + 2 x 2 − 4 x + 3 x 2 + 3 x − 6
Combining Like Terms Next, we combine like terms. We have 2 x 3 , 2 x 2 and 3 x 2 , − 4 x and 3 x , and − 6 . Combining these gives:
2 x 3 + ( 2 x 2 + 3 x 2 ) + ( − 4 x + 3 x ) − 6 2 x 3 + 5 x 2 − x − 6
Final Answer Therefore, the simplified product is 2 x 3 + 5 x 2 − x − 6 .
Examples
Polynomials are used to model curves and shapes in various fields, such as engineering, computer graphics, and economics. For example, engineers might use polynomials to design the curve of a bridge or the shape of a car body. In computer graphics, polynomials are used to create smooth curves and surfaces for 3D models. Economists use polynomials to model cost and revenue functions to analyze market trends and make predictions. Understanding how to multiply and simplify polynomials is essential for these applications.
