# Week 14 – precalculus 11

Equivalent Rational Expressions:

To simplify a Rational Expression, we must factor the numerator and denominator, cancel out any common factors and find the Non-permissible values. We can better understand this by showing it.

If we are given the expression $\frac {x^2 + 8x + 20} {x^2 - 25}$

We will then simplify or factor the top and bottom

$\frac {(x+5)(x+4)} {(x-5)(x+5)}$

Now in a fraction, the denominator cannot equal 0, the numbers we cannot use for x to prevent this are called non-permissible values. These values will be what makes the bottom equal zero, in this case, 5 and -5. So x cannot equal 5 or -5. This will be shown in our final answer.

We will now cancel out common monomials or binomials, in this case x+5, giving us:

$\frac {(x+4)} {(x-5)} x \neq 5,-5$

x cannot equal 5 or -5

The three steps are 1: Factor, 2: Find the Non-permissible values, 3: Cancel out common factors.

Now that we have the basics down I will get into some situations and what to do.

In a situation where you are given $\frac {2x}{12x^2 + 2x}$ one may want to just cancel out 2x but in reality the + or – sign connects the two values on the bottom and therefore you cannot just cancel out one part, you will factor the bottom and then work it out.

Coefficients can be canceled out and anything that is multiplying.

One thing to mention is that if you are give an expression like $\frac {2x}{2x(6x+1)}$ you can actually cancel out the coefficient on the bottom because it is seperate and you will be left with $\frac {1}{(6x+1)}$