A. $\left(15 x^2-2 x+3\right)-\left(5+6 x+7 x^2\right)$

Distribute the negative sign: 15 x 2 − 2 x + 3 − 5 − 6 x − 7 x 2 .
Combine like terms: ( 15 x 2 − 7 x 2 ) + ( − 2 x − 6 x ) + ( 3 − 5 ) .
Simplify: 8 x 2 − 8 x − 2 .
The simplified expression is 8 x 2 − 8 x − 2 ​ .

Explanation

Understanding the problem We are asked to simplify the expression ( 15 x 2 − 2 x + 3 ) − ( 5 + 6 x + 7 x 2 ) . This involves combining like terms after distributing the negative sign.

Distributing the negative sign First, distribute the negative sign in the second parenthesis: 15 x 2 − 2 x + 3 − 5 − 6 x − 7 x 2

Grouping like terms Next, group the like terms together: ( 15 x 2 − 7 x 2 ) + ( − 2 x − 6 x ) + ( 3 − 5 )

Combining like terms Now, combine the like terms by performing the arithmetic operations: ( 15 x 2 − 7 x 2 ) = 8 x 2 ( − 2 x − 6 x ) = − 8 x ( 3 − 5 ) = − 2

Writing the final expression Finally, write the simplified expression: 8 x 2 − 8 x − 2


Examples
Polynomials are used to model curves and shapes in various fields such as engineering, computer graphics, and physics. For example, the trajectory of a projectile can be modeled using a quadratic polynomial. Simplifying polynomial expressions allows engineers to optimize designs, predict behavior, and make informed decisions. Understanding polynomial operations is crucial for solving real-world problems involving curves and approximations.

Answered by GinnyAnswer | 2025-07-08