Problem 62 - Algebra for College Students

Question

For Problems 59-68, simplify each rational expression. You may want to refer to Example 12 of this section. \(\frac{9-y}{y^{2}-81}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-1}{y+9} \).

Step 1: Identify Common Patterns

The expression is \( \frac{9-y}{y^{2}-81} \). Notice that the denominator \( y^{2}-81 \) is a difference of squares, which can be factored.

Step 2: Factor the Denominator

The expression \( y^2 - 81 \) is a difference of squares. Use the identity \( a^2 - b^2 = (a-b)(a+b) \). Here, \( a = y \) and \( b = 9 \). So, \( y^2 - 81 = (y-9)(y+9) \).

Step 3: Rewrite the Numerator

Notice that \( 9 - y \) can be rewritten as \( -(y - 9) \) because \( 9 - y = -1(y - 9) \).

Step 4: Substitute and Simplify

Substitute the factored and rewritten parts back into the expression: \( \frac{9-y}{y^{2}-81} = \frac{-(y-9)}{(y-9)(y+9)} \). The \( y-9 \) terms cancel out.

Step 5: Final Simplified Expression

After canceling \( y-9 \), the simplified version of the rational expression is \( \frac{-1}{y+9} \).

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra, which involves breaking down polynomial expressions into simpler, multiplicative factors. By recognizing patterns such as the difference of squares, we can simplify expressions efficiently.

A difference of squares is a special pattern characterized by an expression in the form of

• \(a^2 - b^2 = (a-b)(a+b)\)

This means that the polynomial can be factored into two binomials. For example, in the expression \(y^2 - 81\), \(y^2\) and \(81\) are perfect squares, making this polynomial a candidate for factoring into \((y-9)(y+9)\).

Recognizing such patterns enables transforming complex expressions into simpler ones, aiding in further simplification by reducing complicated calculations.

Simplifying Expressions

Simplifying expressions involves reducing a given mathematical expression into its most concise form, where no further simplification is possible. It requires utilizing mathematical identities and cancelling out terms where applicable.

In rational expressions, simplification often involves cancelling terms that appear in both the numerator and the denominator. This emerges when you successfully factor polynomials and rewrite the expressions properly.

For instance, in the expression \(\frac{9-y}{y^{2}-81}\), following factoring, rewrite \(9-y\) as \(-(y-9)\). This allows the \(y-9\) terms in both the numerator and denominator to cancel each other out, leading to the simplified expression \(\frac{-1}{y+9}\).

Simplifying makes expressions easier to interpret, often simplifying further calculations or applications within problems.

Algebraic Techniques

Algebraic techniques refer to various methods and strategies used to manipulate and simplify algebraic expressions. These include factoring, identifying common patterns, and applying arithmetic operations effectively. Understanding and using these techniques helps in tackling a broad range of algebraic problems.

One common technique is factorization, which breaks down expressions into simpler components, assisting in identifying and canceling common factors across numerators and denominators.

Another technique is rewriting expressions by recognizing the structure and identity properties. For example, the expression \(9-y\) can be seen as \(-(y-9)\) due to the distributive property of multiplication over subtraction.

Mastering these techniques ensures a more manageable approach to solving algebraic equations and expressions. This way, students gain confidence in moving from more complicated algebra problems to generating clean, precise solutions.