Subtract -9x^2y^2+5x^2y-11xy+1 4xy-13x^2y+7
The question is asking to perform the operation of subtraction between two algebraic expressions. The first expression is "-9x^2y^2 + 5x^2y - 11xy + 1" and the second expression is "4xy - 13x^2y + 7". You are required to subtract each term of the second expression from the corresponding term of the first expression, taking care to properly combine like terms—that is, terms that have the same variables raised to the same powers—and take into account the rules of subtracting negative numbers.
$- 9 x^{2} y^{2} + 5 x^{2} y - 11 x y + 1$$4 x y - 13 x^{2} y + 7$
Answer
Solution:
Step 1:
Begin by subtracting the polynomial $4xy - 13x^2y + 7$ from $-9x^2y^2 + 5x^2y - 11xy + 1$. Write the expression as follows:
$$-9x^2y^2 + 5x^2y - 11xy + 1 - (4xy - 13x^2y + 7)$$
Step 2:
Proceed to simplify the expression.
Step 2.1:
Distribute the negative sign across the second polynomial.
$$-9x^2y^2 + 5x^2y - 11xy + 1 - 4xy + 13x^2y - 7$$
Step 2.2:
Perform the simplification of terms.
Step 2.2.1:
Multiply $4$ by $-1$ to get $-4xy$.
$$-9x^2y^2 + 5x^2y - 11xy + 1 - 4xy + 13x^2y - 7$$
Step 2.2.2:
Multiply $-13$ by $-1$ to get $+13x^2y$.
$$-9x^2y^2 + 5x^2y - 11xy + 1 - 4xy + 13x^2y - 7$$
Step 2.2.3:
Multiply $-1$ by $7$ to get $-7$.
$$-9x^2y^2 + 5x^2y - 11xy + 1 - 4xy + 13x^2y - 7$$
Step 2.3:
Eliminate the parentheses and combine like terms.
$$-9x^2y^2 + (5x^2y + 13x^2y) + (-11xy - 4xy) + (1 - 7)$$
Step 3:
Combine like terms to simplify the expression.
Step 3.1:
Add $5x^2y$ and $13x^2y$.
$$-9x^2y^2 + 18x^2y - 11xy - 4xy + 1 - 7$$
Step 3.2:
Combine $-11xy$ and $-4xy$.
$$-9x^2y^2 + 18x^2y - 15xy + 1 - 7$$
Step 3.3:
Subtract $7$ from $1$.
$$-9x^2y^2 + 18x^2y - 15xy - 6$$
Knowledge Notes:
The problem involves subtracting one polynomial from another. Here are the relevant knowledge points:
Polynomials: An expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Subtraction of Polynomials: To subtract polynomials, you distribute the negative sign to the second polynomial and combine like terms.
Distributive Property: This property is used to remove parentheses when subtracting polynomials: $a(b + c) = ab + ac$ and $-1 \cdot (b - c) = -b + c$.
Combining Like Terms: Terms that have the same variable factors are combined by adding or subtracting their coefficients.
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.
