Subtract -9x^2y^2+5x^2y-11xy+1 4xy-13x^2y+7

Solution:

Step 1:

Begin by subtracting the polynomial $4xy-13x^{2}y+7$ from $-9x^2{y}^2+5x^{2}y-11xy+1$. Write the expression as follows:

$$-9x^2{y}^2+5x^{2}y-11xy+1-(4xy-13x^{2}y+7)$$

Step 2:

Proceed to simplify the expression.

Step 2.1:

Distribute the negative sign across the second polynomial.

$$-9x^2{y}^2+5x^{2}y-11xy+1-4xy+13x^{2}y-7$$

Step 2.2:

Perform the simplification of terms.

Step 2.2.1:

Multiply $4$ by $-1$ to get $-4xy$.

$$-9x^2{y}^2+5x^{2}y-11xy+1-4xy+13x^{2}y-7$$

Step 2.2.2:

Multiply $-13$ by $-1$ to get $+13x^{2}y$.

$$-9x^2{y}^2+5x^{2}y-11xy+1-4xy+13x^{2}y-7$$

Step 2.2.3:

Multiply $-1$ by $7$ to get $-7$.

$$-9x^2{y}^2+5x^{2}y-11xy+1-4xy+13x^{2}y-7$$

Step 2.3:

Eliminate the parentheses and combine like terms.

$$-9x^2{y}^2+(5x^{2}y+13x^{2}y)+(-11xy-4xy)+(1-7)$$

Step 3:

Combine like terms to simplify the expression.

Step 3.1:

Add $5x^{2}y$ and $13x^{2}y$.

$$-9x^2{y}^2+18x^{2}y-11xy-4xy+1-7$$

Step 3.2:

Combine $-11xy$ and $-4xy$.

$$-9x^2{y}^2+18x^{2}y-15xy+1-7$$

Step 3.3:

Subtract $7$ from $1$.

$$-9x^2{y}^2+18x^{2}y-15xy-6$$

Knowledge Notes:

The problem involves subtracting one polynomial from another. Here are the relevant knowledge points:

1. Polynomials: An expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

2. Subtraction of Polynomials: To subtract polynomials, you distribute the negative sign to the second polynomial and combine like terms.

3. Distributive Property: This property is used to remove parentheses when subtracting polynomials: $a(b+c)=ab+ac$ and $-1\cdot (b-c)=-b+c$.

4. Combining Like Terms: Terms that have the same variable factors are combined by adding or subtracting their coefficients.

5. Simplification: The process of combining like terms and performing arithmetic operations to reduce a polynomial to its simplest form.

In LaTeX, the caret symbol (^) indicates an exponent, and the underscore (_) is used for subscripts. The use of parentheses is crucial for maintaining the correct order of operations, and the negative sign must be distributed correctly when subtracting polynomials.