Problem 54 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ (x-2 y)^{2}+2(x+y)(x-3 y)+x(2 x+3 y+2) $$

Step-by-Step Solution

Answer

The simplified expression is \(5x^2 - 5xy - 2y^2 + 2x\).

Step 1: Expanding the squares and products

First, expand the given expression: \((x-2y)^2 + 2(x+y)(x-3y) + x(2x+3y+2)\) Expanding the square and products, we get: \(x^2 - 4xy + 4y^2 + 2x^2 - 6xy + 2xy - 6y^2 + 2x^2 + 3xy + 2x\)

Step 2: Combining like terms

Now, let's combine terms that are alike: \(x^2 + 2x^2 + 2x^2 - 4xy - 6xy + 2xy + 3xy + 4y^2 - 6y^2 + 2x\)

Step 3: Simplifying the expression

Finally, let's simplify the expression by adding and subtracting terms: \(5x^2 - 5xy - 2y^2 + 2x\) This is the simplified expression.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They can represent real-world situations and are often used to formulate mathematical problems. For example, the expression \((x - 2y)^2 + 2(x+y)(x-3y) + x(2x+3y+2)\) consists of terms involving variables \(x\) and \(y\), numbers, and operations like addition and multiplication. It's an abstract representation that can be manipulated and simplified using mathematical rules.
Understanding algebraic expressions involves recognizing their components:

• **Terms**: These are individual parts of an expression, separated by `+` or `-` signs.
• **Coefficients**: Numbers multiplying variables. In `5x`, the coefficient is 5.
• **Variables**: Symbols representing numbers, usually letters like \(x\) and \(y\).
• **Constants**: Fixed numbers without variables, like 2 or -3.

Learning to identify these components is crucial for working with algebraic expressions effectively. It allows you to apply different operations and simplify them correctly.

Polynomial Operations

Polynomial operations involve arithmetic actions that are performed on polynomials, such as addition, subtraction, multiplication, and division. These operations are essential for solving algebraic equations and simplifying expressions.
When you see a problem like \((x-2y)^2 + 2(x+y)(x-3y) + x(2x+3y+2)\), you need to use polynomial operations to expand and simplify it.

• **Addition and Subtraction**: Combine like terms by adding or subtracting their coefficients. Like terms have the same variables and exponents.
• **Multiplication**: Use rules like distributing each term and applying the FOIL method for binomials. For the term \((x-2y)^2\), multiply \((x-2y)\) by itself.
• **Expanding Terms**: Distribute each term in a product across other terms in a grouping (like parentheses). This operation lays the groundwork for further simplification.

Mastering these polynomial operations is key to working with more complex algebraic expressions. They enable you to transform an expression into its simplest form, making it easier to interpret and use in solving equations.

Mathematical Simplification

Mathematical simplification is the process of reducing an expression to its simplest form by performing operations and combining like terms. It involves making an expression as concise as possible, without altering its value.
To simplify the expression \((x-2y)^2 + 2(x+y)(x-3y) + x(2x+3y+2)\), we:

• **Expand Each Term**: First, rewrite the squared terms and products with their expanded forms, breaking them down into individual terms.
• **Combine Like Terms**: Next, collect terms with the same variable parts. For instance, \(-4xy + 2xy + 3xy\) can be combined to \(-5xy\).
• **Reorder and Simplify**: Group similar types of terms and perform arithmetic operations to further condense the expression.

Ultimately, the aim of simplification is to present an expression in its most understandable and basic form. In the solution provided, the final outcome \(5x^2 - 5xy - 2y^2 + 2x\) is achieved after correctly simplifying the given expression. Simplified expressions are easier to work with, especially when substitution and further calculations are necessary.