Problem 68 - Algebra 2

Question

Simplify each rational expression. $$ \frac{c^{2}-8 c+12}{c^{2}-11 c+30} $$

Step-by-Step Solution

Answer

The simplified form of the rational expression \(\frac{c^{2}-8 c+12}{c^{2}-11 c+30}\) is \(\frac{c-2}{c-5}\).

Step 1: Identify the quadratic polynomials

The rational expression contains the numerator \(c^{2}-8 c+12\) and the denominator \(c^{2}-11 c+30\). Both of these expressions are quadratic polynomials that can be factored to simplify this expression.

Step 2: Factor the numerator

The numerator \(c^{2}-8 c+12\) can be factored into \((c-2)(c-6)\) by the rule of factoring quadratic expressions.

Step 3: Factor the denominator

The denominator \(c^{2}-11 c+30\) can be factored into \((c-5)(c-6)\) by the rule of factoring quadratic expressions.

Step 4: Cancel out common terms

In the rational expression \(\frac{(c-2)(c-6)}{(c-5)(c-6)}\), you can cancel out the common term \((c-6)\) in both the numerator and the denominator to simplify the rational expression to \(\frac{(c-2)}{(c-5)}\).

Key Concepts

Factoring Quadratic Polynomials

Quadratic polynomials often take the form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. To factor them, look for two numbers that multiply to \( a \times c \) and add to \( b \). These numbers help in splitting the middle term when \( a = 1 \), simplifying the expression into products of binomials.
In our example:

• The numerator \( c^{2} - 8c + 12 \) needs to be factored. Look for two numbers that multiply to 12 (the constant term) and add up to -8 (the coefficient of \( c \)). Those numbers are -2 and -6.
• This gives us \( (c-2)(c-6) \) as the factorization of the numerator.
• For the denominator \( c^{2} - 11c + 30 \), search for two numbers that multiply to 30 and add up to -11. These are -5 and -6.
• The denominator can thus be factored into \( (c-5)(c-6) \).

Factoring is an essential first step as it allows for simplification by revealing common factors that can later be canceled.

Canceling Common Factors

Once you factor both the numerator and the denominator of a rational expression, you might find common factors. These are terms that appear in both the top and bottom of the fraction.
The essence of simplifying rational expressions lies in canceling these common factors:

• In the factored rational expression \( \frac{(c-2)(c-6)}{(c-5)(c-6)} \), notice that \( (c-6) \) is a common factor.
• You "cancel" \( (c-6) \) because dividing a number or expression by itself equals 1.
• After canceling, you're left with \( \frac{(c-2)}{(c-5)} \).

This action not only simplifies the expression but also provides a clearer insight into the roots and behavior of the original rational expression.

Rational Expressions

Rational expressions are fractions composed of polynomials in both the numerator and the denominator. They are called "rational" because they resemble rational numbers, which are fractions of integers.
Key aspects of working with rational expressions include:

• Identifying opportunities to simplify by factoring.
• Understanding that any zero of the denominator results in a restriction or undefined point for the expression.
• Maintaining equivalence; simplifying by canceling common factors doesn't change the value of the expression, except at the restrictions.

For the expression \( \frac{c^{2}-8c+12}{c^{2}-11c+30} \), simplifying it to \( \frac{(c-2)}{(c-5)} \) highlights how a seemingly complex fraction can be reduced, given correct factoring and canceling of common factors.