Simplify the following polynomial expression:

$\left(5 x^3+4 x^2+6 x-3\right)+\left(-2 x^3+3 x^2-7 x+9\right)-(5 x+3)\left(4 x^2+x-1\right)$

First, expand the product of the last term: ( 5 x + 3 ) ( 4 x 2 + x − 1 ) = 20 x 3 + 17 x 2 − 2 x − 3 .
Combine like terms in the first two parentheses: ( 5 x 3 + 4 x 2 + 6 x − 3 ) + ( − 2 x 3 + 3 x 2 − 7 x + 9 ) = 3 x 3 + 7 x 2 − x + 6 .
Subtract the expanded product from the combined polynomial: ( 3 x 3 + 7 x 2 − x + 6 ) − ( 20 x 3 + 17 x 2 − 2 x − 3 ) = − 17 x 3 − 10 x 2 + x + 9 .
The simplified polynomial expression is − 17 x 3 − 10 x 2 + x + 9 ​ .

Explanation

Understanding the Problem We are asked to simplify the polynomial expression ( 5 x 3 + 4 x 2 + 6 x − 3 ) + ( − 2 x 3 + 3 x 2 − 7 x + 9 ) − ( 5 x + 3 ) ( 4 x 2 + x − 1 ) . Let's break this down step by step.

Expanding the Product First, we need to expand the product ( 5 x + 3 ) ( 4 x 2 + x − 1 ) . We can do this by multiplying each term in the first parenthesis by each term in the second parenthesis:


( 5 x + 3 ) ( 4 x 2 + x − 1 ) = 5 x ( 4 x 2 + x − 1 ) + 3 ( 4 x 2 + x − 1 )
= 20 x 3 + 5 x 2 − 5 x + 12 x 2 + 3 x − 3
= 20 x 3 + ( 5 x 2 + 12 x 2 ) + ( − 5 x + 3 x ) − 3
= 20 x 3 + 17 x 2 − 2 x − 3

Combining Like Terms Next, we combine like terms in the first two sets of parentheses:

( 5 x 3 + 4 x 2 + 6 x − 3 ) + ( − 2 x 3 + 3 x 2 − 7 x + 9 ) = ( 5 x 3 − 2 x 3 ) + ( 4 x 2 + 3 x 2 ) + ( 6 x − 7 x ) + ( − 3 + 9 )
= 3 x 3 + 7 x 2 − x + 6

Subtracting and Simplifying Now, we subtract the expanded product from the combined polynomial:

( 3 x 3 + 7 x 2 − x + 6 ) − ( 20 x 3 + 17 x 2 − 2 x − 3 ) = 3 x 3 + 7 x 2 − x + 6 − 20 x 3 − 17 x 2 + 2 x + 3
= ( 3 x 3 − 20 x 3 ) + ( 7 x 2 − 17 x 2 ) + ( − x + 2 x ) + ( 6 + 3 )
= − 17 x 3 − 10 x 2 + x + 9

Final Answer Therefore, the simplified polynomial expression is − 17 x 3 − 10 x 2 + x + 9 .

Examples
Polynomials are used to model curves and shapes in engineering and computer graphics. Simplifying polynomial expressions allows engineers to optimize designs, reduce computational complexity, and improve the efficiency of algorithms used in various applications, such as image processing and simulations. For example, in structural engineering, polynomials can represent the stress distribution within a beam, and simplifying these polynomials helps engineers to quickly assess the beam's stability under different loads.

Answered by GinnyAnswer | 2025-07-03

The simplified polynomial expression is − 17 x 3 − 10 x 2 + x + 9 . To arrive at this, we expanded the product, combined like terms, and then performed the subtraction step. Following these steps leads to the final simplified form of the original expression.
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Answered by Anonymous | 2025-07-04