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Chapter 5: Problem 45

Subtract. $$ (2 x 2 y 2-4 x y+9)-(3 x 2 y 2-3 x y-5) $$

Short Answer

Expert verified
The result is \(-x^2y^2 - xy + 14\).

Step by step solution

01

Distribute Negative Sign

Rewrite the subtraction as the addition of the negative of the second expression. To do this, distribute the negative sign across the second polynomial:\[(2x^2y^2 - 4xy + 9) - (3x^2y^2 - 3xy - 5) = (2x^2y^2 - 4xy + 9) + (-3x^2y^2 + 3xy + 5)\]
02

Combine Like Terms

Combine the like terms from both polynomials. Like terms have the same variables raised to the same power:- Combine the \(x^2y^2\) terms: \(2x^2y^2 - 3x^2y^2 = -x^2y^2\)- Combine the \(xy\) terms: \(-4xy + 3xy = -xy\)- Combine the constant terms: \(9 + 5 = 14\).Thus, the expression becomes:\[-x^2y^2 - xy + 14\]
03

Simplify the Expression

Write the final simplified expression from the combined like terms:\[-x^2y^2 - xy + 14\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental principle in algebra that allows us to simplify expressions by distributing a factor across terms inside parentheses. In the context of polynomial subtraction, this involves distributing a negative sign across the terms of a polynomial.
When we have an expression like \[(2x^2y^2 - 4xy + 9) - (3x^2y^2 - 3xy - 5)\]we effectively want to subtract every term in the second polynomial from the terms in the first polynomial.
To achieve this, apply the distributive property to the negative sign placed before the second polynomial:
  • Multiply -1 by each term inside the parentheses: \[-(3x^2y^2 - 3xy - 5) = -3x^2y^2 + 3xy + 5\]
This changes the subtraction into addition and makes it easier to combine the terms from both polynomials.
Like Terms
Like terms in algebra are terms that have the same variables raised to the same powers. When simplifying expressions, it is crucial to identify and combine like terms to achieve a more straightforward expression. For example, in polynomial expressions like:\[2x^2y^2 - 3x^2y^2 \]Both terms are like terms because they share the same variables \(x^2y^2\).
When you see a subtraction or addition operation involving these terms, they can be combined easily:
  • Subtract the coefficients: \[2 - 3 = -1\]
Thus, \[-3x^2y^2 + 2x^2y^2\] simplifies to \[-x^2y^2\].
Recognizing like terms allows simplification by focusing on coefficients while maintaining the integrity of the variable components.
Simplifying Expressions
Simplifying expressions involves combining like terms, distributing any factors or coefficients, and reducing the expression to its simplest form. In this example, after identifying like terms, we move on to combining them to simplify the overall expression. After applying the distributive property, our expression becomes:\[2x^2y^2 - 4xy + 9 + (-3x^2y^2 + 3xy + 5)\]By combining the like terms step by step, we achieve an easier-to-read form:
  • Combine \(x^2y^2\) terms: \[-x^2y^2\]
  • Combine \(xy\) terms: \[-xy\]
  • Combine constant terms: \[^9 + 5 = 14\]
The final simplified expression is:\[-x^2y^2 - xy + 14\].Each term is organized so that the same types of terms are grouped together, making the expression simpler and more efficient to use.

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