Understanding the Algebraic Simplification Problem

The question asks us to simplify the given algebraic expression: \(\frac{x^4 - 2x^2 + 1}{x^2 - 2x + 1}\). To simplify this fraction, we need to factorize both the numerator and the denominator.

Step-by-Step Simplification Process

Step 1: Factorize the Numerator

The numerator is \(x^4 - 2x^2 + 1\). This expression looks like a quadratic form if we consider \(x^2\) as a single variable. Let \(y = x^2\). The expression becomes \(y^2 - 2y + 1\). This is a perfect square trinomial, which factors as \((y-1)^2\).

Now, substitute back \(y = x^2\): \((x^2 - 1)^2\).

The term \(x^2 - 1\) is a difference of squares, which factors as \((x-1)(x+1)\).

So, the numerator becomes \(((x-1)(x+1))^2 = (x-1)^2 (x+1)^2\).

Let's summarize the factorization of the numerator:

• \(x^4 - 2x^2 + 1\)
• Let \(y = x^2\), expression becomes \(y^2 - 2y + 1\)
• Factorize as \((y-1)^2\)
• Substitute back \(y = x^2\): \((x^2 - 1)^2\)
• Factor \(x^2 - 1\) as \((x-1)(x+1)\)
• Numerator fully factored: \((x-1)^2 (x+1)^2\)

Step 2: Factorize the Denominator

The denominator is \(x^2 - 2x + 1\). This is a perfect square trinomial, which factors as \((x-1)^2\).

Let's summarize the factorization of the denominator:

• \(x^2 - 2x + 1\)
• Factorize as \((x-1)^2\)

Step 3: Simplify the Fraction by Cancelling Common Factors

Now substitute the factored forms back into the original expression:

\(\frac{(x-1)^2 (x+1)^2}{(x-1)^2}\)

Assuming \(x \neq 1\) (to avoid division by zero), we can cancel the common factor \((x-1)^2\) from the numerator and the denominator.

The expression simplifies to \((x+1)^2\).

Step 4: Expand the Simplified Expression

Finally, expand the term \((x+1)^2\):

\((x+1)^2 = x^2 + 2(x)(1) + 1^2 = x^2 + 2x + 1\).

So, the simplified expression is \(x^2 + 2x + 1\).

Conclusion and Verification

The simplified form of the expression \(\frac{x^4 - 2x^2 + 1}{x^2 - 2x + 1}\) is \(x^2 + 2x + 1\).

Let's compare this with the given options:

• Option 1: \(x^2 - 2x + 1\)
• Option 2: \(x^2 + 2x + 2\)
• Option 3: \(x^2 + 2x + 1\)
• Option 4: \(x^2 + x + 1\)

The simplified expression \(x^2 + 2x + 1\) matches Option 3.

Revision Table: Key Algebraic Factorizations

Additional Information on Simplifying Rational Expressions

Simplifying rational expressions involves reducing the fraction to its lowest terms. This is done by factoring the numerator and the denominator completely and then cancelling out any common factors.

• Factorization is Key: The ability to factor polynomials correctly is fundamental. Techniques like finding common factors, difference of squares, sum/difference of cubes, and factoring trinomials (like \(ax^2+bx+c\)) are essential.
• Identifying Common Factors: Once factored, look for identical factors in the numerator and denominator.
• Cancellation: Cancel out the common factors. Remember that cancellation is valid only when the factor is non-zero. For expressions involving variables, this typically implies restrictions on the variable (e.g., \(x \neq 1\) in this problem).
• Domain Restrictions: The simplified expression is equivalent to the original expression only for values of the variable where the original expression is defined. In this case, the original expression is undefined when \(x^2 - 2x + 1 = 0\), which is \((x-1)^2 = 0\), meaning \(x=1\). Therefore, the simplification holds for all \(x \neq 1\).
• Final Form: After cancellation, the remaining expression is the simplified form. It might be left in factored form or expanded form, depending on the requirement.