Problem 26 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. $$ \begin{array}{l} 3\left(x^{2} y^{2}+x y-2\right)-5\left(x^{2} y^{2}+6 x y-2\right) \\ +2\left(x^{2} y^{2}+6 x y-2\right) \end{array} $$

Step-by-Step Solution

Answer

The short answer for the given expression is: \(-15xy\).

Step 1: Distribute Constants

First, we will distribute the constants to the terms inside the parentheses. \(3(x^2y^2 + xy - 2) - 5(x^2y^2 + 6xy - 2) + 2(x^2y^2 + 6xy - 2)\)

Step 2: Multiplication

Now, we will multiply each term by the constants: \(3x^2y^2 + 3xy - 6 - 5x^2y^2 - 30xy + 10 + 2x^2y^2 + 12xy - 4\)

Step 3: Combine Like Terms

Combine like terms in the expression: \( (3x^2y^2 - 5x^2y^2 + 2x^2y^2) + (3xy - 30xy + 12xy) + (-6 + 10 - 4)\)

Step 4: Simplify

Simplify each term and rewrite the final expression: \(0x^2y^2 - 15xy + 0\) And the simplified expression is: \(-15xy\)

Key Concepts

Distributive Property

The distributive property is a fundamental principle used in algebra to simplify expressions. It allows you to break down expressions into smaller, more manageable parts. The property is expressed mathematically as:

• \( a(b + c) = ab + ac \)

In practice, this means that you multiply each term inside the parentheses by the factor outside the parentheses. In our exercise, the distributive property helps to simplify terms like \(3(x^2y^2 + xy - 2)\). Each term inside the parentheses gets multiplied by 3:

• \(3 \times x^2y^2\) becomes \(3x^2y^2\)
• \(3 \times xy\) becomes \(3xy\)
• \(3 \times -2\) becomes \(-6\)

By applying this property to each group in the expression, you can transform complex expressions into simpler ones. This process is essential for further steps in algebra, such as combining like terms.

Combining Like Terms

Combining like terms is the process of adding or subtracting terms that have the same variables raised to the same powers. In algebraic expressions, like terms can be summed or subtracted to simplify the expression. Consider this key example from the exercise:

• The terms \(3x^2y^2\), \(-5x^2y^2\), and \(2x^2y^2\) are like terms because they all contain \(x^2y^2\). Combining them results in \((3 - 5 + 2)x^2y^2\), which equals \(0x^2y^2\).
• Similarly, the xy terms \(3xy\), \(-30xy\), and \(12xy\) are combined to \((3 - 30 + 12)xy\), simplifying to \(-15xy\).

The process of combining like terms helps to reduce expressions, making them easier to interpret and work with. This is an important part of simplifying polynomials and solving equations in algebra.

Polynomial Simplification

Polynomial simplification is the process of reducing a polynomial expression into its simplest form. This involves applying properties like distributive property and combining like terms to reduce the expression to a more convenient form.

• Start by distributing the constants through the terms as shown in the steps with expressions like \(3(x^2y^2 + xy - 2)\).
• Next, identify and combine like terms, grouping together similar variables and constants.
• Finally, perform any necessary arithmetic operations to arrive at the simplest form, as demonstrated in the step-by-step solution, which results in the simplified expression \(-15xy\).

Simplifying polynomials is crucial, as it makes further operations like addition, subtraction, or even solving equations much easier. Mastering these steps ensures a better understanding of algebraic expressions and aids in solving more complex problems efficiently.