Lesson Objectives
• Demonstrate an understanding of how to factor polynomials
• Learn how to find the restricted values for a rational expression

How to Find the Domain of a Rational Expression

A rational number is any number that can be written as the quotient of two integers, with a non-zero denominator. Some examples: $$\frac{2}{3}, \frac{-8}{7}, -7, 5$$ Since any integer can be written over 1, all integers are also rational numbers.
An algebraic fraction or rational expression is the quotient of two polynomials, with a non-zero denominator. In our textbook, we will see a rational expression defined as: $$\frac{P}{Q}, Q ≠ 0$$ Where P and Q are polynomials.
Since a rational expression is a fraction, we must make sure that we never divide by zero. One of the first tasks we face with rational expressions is finding where the rational expression is undefined. We can say that the domain (set of allowable x-values) of a rational expression/function includes all values that result in a non-zero denominator. In other words, to find the domain, find values for x that make the denominator equal to zero and exclude those values. Let's look at a few examples.
Example 1: Find the excluded values for each, state the domain $$f(x)=\frac{x^2 + 5x - 6}{x^2 + 16x + 60}$$ To find the restricted values, we want to set the denominator equal to zero and solve for x. The solutions will give us the values for x that result in a denominator of zero. These values will represent the restricted values for the rational expression/function. $$x^2 + 16x + 60=0$$ Factor the left side: $$(x + 6)(x + 10)=0$$ Solve using the zero-product property: $$x + 6=0$$ $$x=-6$$ $$x + 10=0$$ $$x=-10$$ This tells us the rational function is not defined for an x-value of -6 or an x-value of -10.
domain:{x| x ≠ -6,-10}
Example 2: Find the excluded values for each, state the domain $$f(x)=\frac{7x^2 - 8x + 1}{14x^3 - 20x^2 + 6x}$$ To find the restricted values, we want to set the denominator equal to zero and solve for x. The solutions will give us the values for x that result in a denominator of zero. These values will represent the restricted values for the rational expression/function. $$14x^3 - 20x^2 + 6x=0$$ Factor the left side: $$2x(7x^2 - 10x + 3)=0$$ $$2x(7x - 3)(x - 1)=0$$ Solve using the zero-product property: $$2x=0$$ $$x=0$$ $$7x - 3=0$$ $$x=\frac{3}{7}$$ $$x - 1=0$$ $$x=1$$ This tells us the rational function is not defined for an x-value of 0, 1, or 3/7.
domain:{x | x ≠ 0, 1, 3/7}

Skills Check:

Example #1

Find the domain. $$\frac{2x^3 + 2x^2 - 12x}{2x^2 + 10x + 12}$$

A
$$\{x | x ≠ -5,7\}$$
B
$$\{x | x ≠ -3,-2\}$$
C
$$\{x | x ≠ -1,0,6\}$$
D
$$\{x | x ≠ 5,2\}$$
E
$$\{x | x ≠ -3,1,7\}$$

Example #2

Find the domain. $$\frac{2x^3 + 6x^2 - 8x}{3x^3 - 9x^2 + 6x}$$

A
$$\{x | x ≠ -4,0,1\}$$
B
$$\{x | x ≠ 2,5\}$$
C
$$\{x | x ≠ 3\}$$
D
$$\{x | x ≠ 0,1,2\}$$
E
$$\{x | x ≠ -5,-1\}$$

Example #3

Find the domain. $$\frac{7x + 42}{3x^2 + 33x + 90}$$

A
$$\left\{x | x ≠ -1,-\frac{4}{5}\right\}$$
B
$$\{x | x ≠ -5,9\}$$
C
$$\{x | x ≠ -3,0,10\}$$
D
$$\{x | x ≠ -2,0\}$$
E
$$\{x | x ≠ -6,-5\}$$