Problem 90 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{y^{2}-2 y+1}{1-y^{4}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1-y}{(1+y)(1+y^2)} \).

Step 1: Observe the Expression

The given expression is \( \frac{y^{2}-2y+1}{1-y^{4}} \). Notice that the numerator and denominator may be factorable.

Step 2: Factor the Numerator

The numerator \( y^2 - 2y + 1 \) can be expressed as \( (y-1)^2 \) since it is a perfect square trinomial.

Step 3: Factor the Denominator

The denominator \( 1 - y^4 \) can be rewritten as \( (1-y^2)(1+y^2) \) because it is a difference of squares. Further factor \( 1-y^2 \) into \( (1-y)(1+y) \), resulting in \( (1-y)(1+y)(1+y^2) \).

Step 4: Simplify the Expression

Substitute the factored forms into the original expression: \( \frac{(y-1)^2}{(1-y)(1+y)(1+y^2)} \). Recall that \( 1-y = -(y-1) \), thus the denominator can be rewritten as \( -(y-1)(1+y)(1+y^2) \).

Step 5: Cancel Common Factors

The expression \( \frac{(y-1)^2}{-(y-1)(1+y)(1+y^2)} \) simplifies by canceling \( y-1 \), resulting in \( \frac{y-1}{-(1+y)(1+y^2)} \). After canceling, the remaining factor \( y-1 \) gives: \( \frac{-(y-1)}{(1+y)(1+y^2)} \).

Step 6: Simplify the Sign

The negative sign in the denominator can be moved to the numerator: \( -\frac{y-1}{(1+y)(1+y^2)} \). This yields the simplified expression \( \frac{1-y}{(1+y)(1+y^2)} \).

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that helps to simplify complex expressions. It's similar to breaking down a whole number into its prime factors. For polynomials, it involves expressing them as a product of simpler polynomials. This simplification process is especially helpful when trying to reduce fractions. Let's take an example. Consider the polynomial \( y^2 - 2y + 1 \). To factor this, identify the pattern or use techniques such as trial and error or recognizing perfect square trinomials. - A perfect square trinomial has a specific form: \((a-b)^2 = a^2 - 2ab + b^2\). - In this case, \( y^2 - 2y + 1 \) matches the form because \( a = y \) and \( b = 1 \), thus it factors to \((y-1)^2\). Recognizing these patterns allows you to simplify complex expressions more easily, a crucial step when working with rational expressions.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials, similar to algebraic fractions. Simplifying rational expressions involves factoring both the numerator and the denominator and then canceling out the common factors. For instance, consider the expression \( \frac{y^2 - 2y + 1}{1-y^4} \). - Factor the numerator as \((y-1)^2\), a perfect square. - The denominator involves recognizing a difference of squares, which can be factored further. After obtaining the factored form, any common terms in the numerator and denominator can be canceled, but only after ensuring they do not render the expression undefined. It’s important to note restrictions on the variable to avoid dividing by zero. Simplifying rational expressions is vital in many areas of algebra as it prepares the groundwork for solving and analyzing more complex algebraic equations.

Difference of Squares

The difference of squares is a specific algebraic identity used to factor polynomials of the form \( a^2 - b^2 \). This identity is fundamental when simplifying rational expressions and other algebraic forms.The difference of squares can be factored as: - \( a^2 - b^2 = (a-b)(a+b) \).In the context of the given problem, \( 1-y^4 \) is a difference of squares: - Recognize that \( y^4 = (y^2)^2 \). Therefore, \( 1-y^4 \) can be rewritten as \( (1-y^2)(1+y^2) \). Further factor \( 1-y^2 \) as \( (1-y)(1+y) \), using the difference of squares identity again.Applying this identity effectively breaks down complicated expressions into simpler, more manageable pieces. Mastering the difference of squares is crucial for students as it pops up frequently in diverse mathematical scenarios, from simple problems to more advanced applications in higher-level mathematics.