We want to write a rational expression that has 4 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 4 and - 3 the excluded values of a rational expression, we have to find a polynomial which is 0 when x= 4 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= 4 and x= - 3, respectively.
a
x-a
Simplify
4
x- 4
x-4
- 3
x-( - 3)
x+3
A polynomial obtained by multiplying our two binomials, x-4 and x+3, will be 0 when x= 4 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-4)(x+3).
The polynomial x^2-x-12 will be the denominator of our rational expression.
Writing the Rational Expression
We have already found the denominator of our rational expression.
â– /x^2-x-12
We can use any polynomial as the numerator. The most basic polynomial is a real number, so let's use 1 as the numerator of our rational expression.
1/x^2-x-12
The excluded values of the above rational expression are 4 and - 3. Note that there are infinitely many rational expressions that have 4 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.
Change
Simplify
5(1/x^2-x-12)
5/5x^2-5x-60
x^2+2/x^2-x-12
x^2+2/x^2-x-12
1/x(x^2-x-12)
1/x^3-x^2-12x
x+1/3(x^2-x-12)
x+1/3x^2-3x-36
All of the above expressions have 4 and - 3 as excluded values as well.
