Problem 64 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ \left(3 x^{2} y-6 x y+x^{2} y^{2}-5\right)-\left(11 x^{2} y^{2}-1+5 y x^{2}\right) $$

Step-by-Step Solution

Answer

\( -10x^2y^2 - 2x^2y - 6xy - 4 \).

Step 1: Distribute the negative sign

\( (3x^2y - 6xy + x^2y^2 - 5) - (11x^2y^2 - 1 + 5x^2y) \)
\( = 3x^2y - 6xy + x^2y^2 - 5 - 11x^2y^2 + 1 - 5x^2y \).

Step 2: Combine like terms

\( x^2y^2 \) terms: \( 1 - 11 = -10x^2y^2 \)
\( x^2y \) terms: \( 3 - 5 = -2x^2y \)
\( xy \) terms: \( -6xy \)
Constants: \( -5 + 1 = -4 \)

Step 3: State the result

\( -10x^2y^2 - 2x^2y - 6xy - 4 \).

Key Concepts

Distributive Property

The distributive property is a useful tool for handling situations where a single term affects multiple terms within a set of parentheses. Specifically, in subtraction of polynomials, it involves distributing a negative sign across terms. Think of it as a simple rule that helps in breaking down complex expressions into simpler parts.

When you have an expression like \(-a(b + c)\), the negative sign must be distributed to both \(b\) and \(c\), changing their signs. This turns \(-a(b + c)\) into \(-ab - ac\).

• Apply the negative sign to each term in the polynomial inside the parentheses.
• The operation changes from \(+\) to \(-\) and vice versa, for each term impacted by the negative.

This step is critical before moving on to other operations, ensuring accuracy in subtraction. Remember, the accuracy in distributing the sign will influence the entire solution.

In the exercise provided: After distributing the negative, the expression became \(3x^2y - 6xy + x^2y^2 - 5 - 11x^2y^2 + 1 - 5yx^2\), aligning all terms and replacing their signs accordingly.

Combining Like Terms

Combining like terms in polynomials is essential to simplify the expressions. Think of it as a process of grouping and adding/subtracting terms that share the same variables and exponents.

Imagine sorting items in your room. You group all books together, clothes in another pile, etc. It's the same with like terms: group terms with matching variables and exponents.

• Find terms with identical variable parts, such as \(x^2y\).
• Add or subtract their coefficients.

Remember that combining like terms simplifies the polynomial and can reveal equivalent expressions. Be careful not to mix up terms which appear similar but differ in their variable parts.

In the given exercise, terms like \(x^2y^2\) and \(x^2y\) are separately combined. Ensure each group operates independently for accurate results.

Subtraction of Polynomials

Subtracting polynomials may sound complicated initially, but becomes manageable through methodical steps. It involves careful attention to signs and terms.

Start by organizing the expression so like terms can be easily identified and matched. This is crucial as the first preparatory step.

• Use the distributive property to handle subtraction across multi-term expressions.
• Carefully subtract like terms ensuring each calculation retains direction (positive or negative).

Subtraction is all about precision, unlike addition where order doesn't impact the outcome due to the commutative property. Be vigilant with sign changes after distributing and during the subtracting process.

In the original problem, you distributed the negative sign converting the second polynomial, then proceed to subtract each like term. By managing this carefully, the solution unfolds systematically.