Problem 52 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ 3 x^{2}-\left\\{x^{2}+1-x[x-(2 x-1)]\right\\}+2 $$

Step-by-Step Solution

Answer

The simplified expression is: \(3x^2 - x + 1\).

Step 1: Expand the expressions inside the brackets.

First we need to expand the expression within the curly brackets. Let's carry out that multiplication: $$ x[x-(2x-1)] = x[x-2x+1] $$ Now let's rewrite the original expression with this expanded expression: $$ 3x^2-\left\\{x^2 + 1 - x[x-2x+1]\right\\}+2 = 3x^2-\left\\{x^2 + 1 - x(x-2x+1)\right\\}+2 $$

Step 2: Simplify the terms within the brackets.

Now, let's simplify the expression inside the brackets: $$ x(x-2x+1) = x(-x+1) = -x^2 + x $$ Now let's rewrite the original expression with this simplified expression: $$ 3x^2-\left\\{x^2 + 1 - x(x-2x+1)\right\\}+2 = 3x^2-\{x^2 + 1 - (-x^2 + x)\}+2 $$

Step 3: Combine the terms from inside and outside the brackets.

Now we'll remove the curly brackets and simplify the expression further: $$ 3x^2-\{x^2 + 1 - (-x^2 + x)\}+2 = 3x^2 - x^2 - 1 + x^2 - x + 2 $$

Step 4: Simplify the final expression.

Finally, let's simplify the expression by combining like terms: $$ 3x^2 - x^2 + x^2 - x + 2 - 1 = (3x^2-x^2+x^2) - x + 1 = 3x^2 - x + 1 $$ Thus, the simplified expression is: $$ 3x^2 - x + 1 $$

Key Concepts

Algebraic Operations

Algebraic operations involve manipulating algebraic expressions to simplify, solve, or rewrite them in a different form. These include basic operations like addition, subtraction, multiplication, and division, as well as more complex processes like factoring and expanding expressions.
In the exercise, the main algebraic operation involved is expanding the brackets using multiplication and combining like terms through addition and subtraction.
When performing algebraic operations, it is crucial to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This ensures that all parts of the expression are simplified correctly and in the proper order.

Expression Expansion

Expression expansion is the process of removing parentheses by multiplying each term inside the parentheses with the term outside. This is essential for simplifying and solving more complex algebraic expressions.
In the solution given, the expression within the brackets, \( x[x-(2x-1)] \), is expanded by multiplying each component inside by \( x \).
Here is the breakdown:

• Multiply \( x \) by each term within: \( x \cdot x, x \cdot (-2x), x \cdot 1 \).
• The resulting expression is \( x^2 - 2x^2 + x \).

Having expanded the terms, you then combine them, allowing further simplification in subsequent steps.

Like Terms

Like terms are terms that have identical variable parts raised to the same power, allowing them to be combined through addition or subtraction.
In polynomial expressions, like terms make simplification possible by combining coefficients while keeping the variable part unchanged.

• In the solution, you see this when similar terms \( 3x^2, -x^2, \) and \( x^2 \) are brought together. Instead of simplifying terms with different variable parts, only the coefficients of like terms are combined.

Recognizing and combining like terms effectively reduces the complexity of expressions, crucial for simplifying large polynomials.

Polynomial Expressions

Polynomial expressions are composed of terms that include variables raised to whole number exponents and coefficients. They can consist of one term, like \( 5x^2 \), or many, as seen in the given exercise.
The key to working with polynomial expressions is understanding their structure and how to manipulate them through operations such as addition, subtraction, and multiplication.
Simplification in polynomials involves:

• Expanding products into sums of terms.
• Identifying and combining like terms.
• Rewriting the expression in a simplified form.

Mastering these manipulations in polynomial expressions allows you to simplify and solve equations with ease, an essential skill in Algebra.