Subtract (7x^2y^2-6x^3+xy)-(5x^2y^2-x^3+xy+x)

Solution:

Step 1: Simplify the polynomial expression

Step 1.1: Distribute the negative sign across the terms in the parentheses

Apply the distributive property to remove the parentheses.
$7x^2{y}^2-6x^3+xy-(5x^2{y}^2)-(-x^3)-(xy)-x$

Step 1.2: Simplify the expression

Step 1.2.1: Apply the negative sign to $5x^2{y}^2$

Multiply $-1$ by $5x^2{y}^2$.
$7x^2{y}^2-6x^3+xy-5(-1)(x^2{y}^2)+x^3-xy-x$

Step 1.2.2: Simplify the double negative on $-x^3$

Step 1.2.2.1: Multiply $-1$ by $-1$

Apply the multiplication of two negatives to get a positive.
$7x^2{y}^2-6x^3+xy-5x^2{y}^2+(-1)(-1)x^3-xy-x$

Step 1.2.2.2: Multiply $x^3$ by $1$

Simplify the expression by multiplying $x^3$ by $1$.
$7x^2{y}^2-6x^3+xy-5x^2{y}^2+x^3-xy-x$

Step 2: Combine like terms

Step 2.1: Group and combine similar terms

Combine terms with the same variables and exponents.
$7x^2{y}^2-5x^2{y}^2-6x^3+x^3+xy-xy-x$

Step 2.1.1: Cancel out $xy-xy$

Subtract $xy$ from $xy$ to eliminate those terms.
$7x^2{y}^2-5x^2{y}^2-6x^3+x^3-x$

Step 2.1.2: Combine the terms

Add and subtract the coefficients of like terms.
$2x^2{y}^2-5x^3-x$

Step 2.2: Subtract $5x^2{y}^2$ from $7x^2{y}^2$

Combine the $x^2{y}^2$ terms.
$2x^2{y}^2-5x^3-x$

Step 2.3: Add $-6x^3$ and $x^3$

Combine the $x^3$ terms.
$-5x^3+2x^2{y}^2-x$

Knowledge Notes:

To solve the given problem, several mathematical concepts and properties are used:

1. Distributive Property: This property is used to eliminate parentheses by distributing the multiplication over addition or subtraction within the parentheses. For example, $a(b+c)=ab+ac$.

2. Combining Like Terms: Terms that have identical variable parts (same variables raised to the same power) can be combined by adding or subtracting their coefficients. For instance, $3x+2x=5x$.

3. Negative Signs and Subtraction: When subtracting polynomials, it's important to distribute the negative sign across all terms within the parentheses. For example, $-(a+b)=-a-b$.

4. Multiplication of Negatives: The product of two negative numbers is positive. For example, $(-1)(-1)=1$.

5. Simplification: The process of combining like terms and simplifying expressions to their simplest form.

In the given solution, these concepts are applied step by step to simplify the polynomial expression by first distributing the negative sign, then combining like terms, and finally simplifying the expression to its most reduced form.