Problem 87 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(-x^4 + x^2 - 2x + 1\).

Step 1: Identify the Expression Type

The expression \((x-1) + x^2\) and \((x-1) - x^2\) is in the form \((a+b)(a-b)\), which is the difference of squares. Here, \(a = x-1\) and \(b = x^2\).

Step 2: Apply the Difference of Squares Formula

The formula for the difference of squares is \((a+b)(a-b) = a^2 - b^2\). Substitute \(a = x-1\) and \(b = x^2\) into this formula to get: \[((x-1)^2 - (x^2)^2)\].

Step 3: Expand the Expression

Square both \((x-1)\) and \(x^2\). - \((x-1)^2 = x^2 - 2x + 1\) - \((x^2)^2 = x^4\)So, the expression becomes \(x^2 - 2x + 1 - x^4\).

Step 4: Combine Like Terms

Starting with \(x^2 - 2x + 1 - x^4\), rearrange and combine terms to simplify the expression: - Combine terms: \(-x^4 + x^2 - 2x + 1\) This is the simplified expression.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In these expressions, variables such as \(x\) represent unknown or variable quantities. Understanding algebraic expressions forms the foundation of solving equations and learning broader algebraic concepts.

In our exercise, the algebraic expression given is \((x-1) + x^2\) and \((x-1) - x^2\). By recognizing that these two expressions are grouped together and noticing how each one is structured is key to understanding the underlying operation.

• The expression \((x-1)\) is known as a binomial, which means it's an algebraic expression with two terms.
• The complete expression involves two binomials being multiplied together.

Identifying components such as polynomials or binomials within algebraic expressions can help simplify complex mathematical problems.

Exploring Polynomial Expansion

Polynomial expansion is an important algebraic technique used to transform expressions into an expanded form. It's crucial when performing operations such as multiplication and can involve distributing terms within parentheses.

In the given problem, the expressions \((x-1+x^2)\) and \((x-1-x^2)\) are multiplied, which brings us to the difference of squares formula:

• The formula \((a+b)(a-b) = a^2 - b^2\) reveals that when two terms are expressed as such, they result in a simplified form of their squares' difference.
• Here, we identify \(a = x-1\) and \(b = x^2\).

Once these variables are substituted into the difference of squares formula, we arrive at the expression \((x-1)^2 - (x^2)^2\). Expanding these terms further helps us break them into simpler, more manageable parts before simplification.

Simplifying Expressions Effectively

Simplifying expressions involves reducing them to their most basic form. This makes the expression easier to understand and work with.

In the exercise, after substituting the differences of squares formula, expansion involves further simplifications:

• First, calculate \((x-1)^2\), which results in \(x^2 - 2x + 1\).
• Calculate \((x^2)^2\) which results in \(x^4\).

Combining these results, we have \(x^2 - 2x + 1 - x^4\). Lastly, we rearrange these to combine like terms, leading to the simplified expression \(-x^4 + x^2 - 2x + 1\).
Using techniques like identifying like terms and recognizing structures such as the difference of squares helps in reducing complex algebraic problems to basic expressions. Mastery of simplification enhances skills in solving complicated algebraic equations seamlessly.