Exercise 21 Page 719

We want to write a rational expression that has 6 and 3 as excluded values. A rational expression is a fraction whose numerator and denominator are polynomials. Rational Expression:polynomial/polynomial An input value is excluded when it makes the polynomial in the denominator equal to 0. Therefore, to make 6 and 3 the excluded values of a rational expression, we have to find a polynomial which is 0 when x= 6 or x= 3 and use it as the denominator. Let's do it!

Finding the Denominator

Note that a binomial x- a is equal 0 when x= a. Using this logic, we can write two binomials which are equal to 0 when x= 6 and x= 3, respectively.

A polynomial obtained by multiplying our two binomials will be 0 when x= 6 or x= 3. This is true because any number multiplied by 0 is equal to 0. Now, let's find the standard form of (x-6)(x-3).

The polynomial x^2-9x+18 will be the denominator of our rational expression.

Writing the Rational Expression

We have already found the denominator of our rational expression. ■/x^2-9x+18 For the numerator, we can use any polynomial. The most basic polynomial is a monomial and the most basic monomial is a real number, so let's use 1 as the numerator of our rational expression. 1/x^2-9x+18 The excluded values of this rational expression are 6 and 3. Note that there are infinitely many rational expressions that have 6 and 3 as excluded values. Some of the ways that we can obtain other expressions that include these excluded values are by multiplying our answer by a constant, by changing the numerator, or by multiplying the denominator by a polynomial.

All of the above expressions have 6 and 3 as excluded values as well.