Problem 88 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(-x^4 - 3x^2 - 4\).

Step 1: Recognize the Formula

Notice that the expression \( \left(x+(2+x^2)\right)\left(x-(2+x^2)\right) \) can be rewritten as \((a+b)(a-b)\) where \(a = x\) and \(b = 2 + x^2\).

Step 2: Apply Difference of Squares

The difference of squares formula is \((a+b)(a-b) = a^2 - b^2\). Apply this formula to simplify \( \left(x+(2+x^2)\right)\left(x-(2+x^2)\right) \) to \(x^2 - (2 + x^2)^2\).

Step 3: Square the Expression

Calculate \((2 + x^2)^2\) as it is part of the difference of squares solution. Expand it by using the formula \((c+d)^2 = c^2 + 2cd + d^2\), where \(c = 2\) and \(d = x^2\).

Step 4: Expand \((2 + x^2)^2\)

Calculate \((2 + x^2)^2\): \[(2 + x^2)^2 = 2^2 + 2\cdot2\cdot x^2 + (x^2)^2 = 4 + 4x^2 + x^4\].

Step 5: Substitute Back

Substitute \(4 + 4x^2 + x^4\) back into the expression \(x^2 - (2 + x^2)^2\) to get \(x^2 - (4 + 4x^2 + x^4)\).

Step 6: Simplify the Expression

Distribute the negative sign and simplify:\[x^2 - 4 - 4x^2 - x^4 = -x^4 - 3x^2 - 4\].

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, etc. In the context of this exercise, we are dealing with the algebraic expression
\[ \left(x+(2+x^2)\right)\left(x-(2+x^2)\right) \]
This expression involves both addition and subtraction within the parentheses, and multiplication outside the parentheses.
Understanding algebraic expressions involves grasping how parts of an expression relate to each other and how they can be manipulated or simplified.

Polynomial Simplification

Polynomial simplification is about reducing a polynomial expression to its simplest form. The ultimate goal is to make an expression easier to understand or work with.
In this exercise, after identifying the difference of squares, which significantly simplifies the process, we continue by expanding and simplifying further.

• Recognize that simplification involves removing parentheses and like terms.
• Here, we have \[x^2 - (4 + 4x^2 + x^4)\]which simplifies to:\[-x^4 - 3x^2 - 4\]
• Each step involves reducing complex expressions into more manageable parts.

Simplification makes polynomials neat and much easier to handle mathematically.

Expanding Binomials

Expanding binomials is the process of multiplying out expressions like \[ (c + d)^2 \].
The key is using formulas like \[ (c+d)^2 = c^2 + 2cd + d^2 \].

For the binomial \[ (2 + x^2)^2 \], here is how you expand it:

• Square the first term: \[ 2^2 = 4 \]
• Multiply double the product of the two terms: \[ 2 \cdot 2 \cdot x^2 = 4x^2 \]
• Square the second term: \[ (x^2)^2 = x^4 \]
• Combine these results: \[ 4 + 4x^2 + x^4 \]

Expanding binomials is an essential skill in algebra that helps simplify and solve complex algebraic expressions.