1. Fractions and Rational
Expressions
Topic 8.1.1

2. Fractions and Rational Expressions
Topic
8.1.1
Lesson
1.1.1
Fractions and Rational Expressions
California Standard:
12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.
What it means for you:
You’ll find out about the conditions for rational numbers to be defined.
Key words:
rational
numerator
denominator
undefined

3. Fractions and Rational Expressions
Topic
8.1.1
Lesson
1.1.1
Fractions and Rational Expressions
In this Topic you’ll find out about the necessary conditions for rational numbers to be defined.

4. Fractions and Rational Expressions
Topic
8.1.1
Lesson
1.1.1
Fractions and Rational Expressions
Rational Expressions Can Be Written as Fractions
A rational expression is any expression that can be written in the form of a fraction — that means it has a numerator and a denominator.
Examples of rational expressions are: 3 1 3 4 8 9 1
x + 1
2x + 1
x – 1
, , , ,
Rational expressions are written in the form , where q ≠ 0. p q

5. Fractions and Rational Expressions
Lesson
1.1.1
Topic
8.1.1
Fractions and Rational Expressions
An Expression is Undefined if the Denominator is Zero
If the denominator is equal to zero, then the expression is said to be undefined (see Topic 1.3.4).
2x + 1
x – 1
So, for example, is defined whenever x is not equal to –0.5.
If x was equal to –0.5, the denominator would be zero…
(2 × –0.5) + 1 = – = 0
… and the expression would be undefined.

6. This means that the expression is undefined when x = –2.
Topic
8.1.1
Fractions and Rational Expressions
Example 1
Determine the value of x for which the expression is undefined.
x + 2 7
Solution
It is undefined when the denominator x + 2 equals zero.
This means that the expression is undefined when x = –2.
Solution follows…

7. Fractions and Rational Expressions
Topic
8.1.1
Fractions and Rational Expressions
Example 2
Determine the value(s) of x for which the expression is undefined.
x2 – 4 2
Solution
It’s undefined when the denominator x2 – 4 equals zero.
So, solve x2 – 4 = 0 to find the values of x:
x2 – 4 = (x – 2)(x + 2) = 0
x – 2 = 0 or x + 2 = 0
x = 2 or x = –2
Therefore, is undefined when x = ±2.
x2 – 4 2
Solution follows…

8. Fractions and Rational Expressions
Topic
8.1.1
Fractions and Rational Expressions
Example 3
Determine the value(s) of x for which the expression is undefined.
x2 – 8x + 15 7x
Solution
Factor the denominator to give:
(x – 3)(x – 5) 7x
If the denominator equals zero, the expression is undefined.
This happens when either (x – 3) or (x – 5) equals zero.
So, the expression is undefined when x = 3 or x = 5.
Solution follows…

9. Fractions and Rational Expressions
Lesson
1.1.1
Topic
8.1.1
Fractions and Rational Expressions
Guided Practice
Determine the value(s) of the variables that make the following rational expressions undefined.
4y – 1 3
y = 1 4
8 + 4x
3 – 2x
5x – 30
x + 1
x = –2
x = 6
a2 – 2a + 1
a2 + 7a + 12 2
4y2 + 11y – 3 –
4k + 5
k – k2
y = , –3 1 4
a = –4, –3
k = 13, –2
x3 – 9x
x2 + 1
2x + 1
x2 – 3x – 28
2a3 – a2
3a3 – 6a2 – 45a
x = 0, 3, or –3
x = 7, –4
a = 0, 5, –3
Solution follows…

10. Fractions and Rational Expressions
Topic
8.1.1
Fractions and Rational Expressions
Independent Practice
Determine the value(s) of the variables that make the following rational expressions undefined.
x – 4 – x2
3x – 6
y3 + y
3y2 + 9y + 6
m + 5
m2 + 8m +7
x = 2
y = –1, –2
m = –1, –7 5x
6x – x2 – x3
2k – 1
3k3 – 3k2 – 18k
6x2 + 13x + 5
8x3 + 32x2 + 30x 3 2
x = 0, – , or – 5
x = 0, 2, or –3
k = 0, 3, or –2
3x + 1
3x2 – 2x – 1
7. Jane states that the rational expression is defined when x is any real number. Show that Jane is incorrect.
x = – or x = 1 will make the denominator 0, so Jane is incorrect 1 3
Solution follows…

11. Fractions and Rational Expressions
Topic
8.1.1
Fractions and Rational Expressions
Round Up
This Topic about the limitations on rational numbers will help you when you’re dealing with fractions in later Topics.
In Topic you’ll simplify rational expressions to their lowest terms.