91Ó°ÊÓ

To understand how to reduce a rational expression to its lowest terms, we first need to know how to factor quadratics.
A quadratic equation is any equation that can be written in the form: \[ ax^2 + bx + c \],where \( a \), \( b \), and \( c \) are constants.
In the exercise, we saw expressions like \( y^2 - 49 \) and \( 3y^2 - 18y - 21 \).
The numerator \( y^2 - 49 \) is a difference of squares and can be factored easily as: \[ y^2 - 49 = (y + 7)(y - 7) \]
The denominator is more complex. We used factoring by grouping: we found numbers that multiply to \( -63 \) (\( 3 \times -21 \)) and add to \( -18 \).
We rewritten the quadratic and then grouped the terms, finally factoring it as: \[ 3y^2 - 18y - 21 = 3(y + 1)(y - 7) \]

A rational expression is a fraction in which the numerator and/or the denominator are polynomials.
In the given exercise, the rational expression was \( \frac{y^2 - 49}{3y^2 - 18y - 21} \).
To simplify such expressions, we factor both the numerator and the denominator, as seen previously, and then reduce the fraction by canceling out common factors.
It’s important to remember:

• Only like terms can be canceled across the numerator and denominator.
• You need to pay close attention to all factors.

After factoring, our rational expression was: \[ \frac{(y + 7)(y - 7)}{3(y + 1)(y - 7)} \]
From here, we can cancel the common factor \( y - 7 \) present in both the numerator and the denominator.

Once we have factored both the numerator and denominator of a rational expression, the next step is simplifying the fraction.
Simplifying means canceling out common factors that appear in both the numerator and the denominator.
In our simplified rational expression, after canceling flat, we got: \[ \frac{y + 7}{3(y + 1)} \]
Remember:

• Always ensure factors are fully simplified before canceling.
• Double-check for any common factors that may have been missed initially.

This method helps in reducing the complexity of the expressions and makes further calculations simpler.
The goal is to make sure that the rational expression is presented in its lowest terms, resulting in a cleaner and more understandable form.