1 topic 8.1.1 fractions and rational expressions fractions and rational expressions

1

Topic 8.1.1Topic 8.1.1

Fractions and Rational

Expressions

Fractions and Rational

Expressions

2

Lesson

1.1.1

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.

Topic

8.1.1

Key words:• rational• numerator• denominator• undefined

Fractions and Rational ExpressionsFractions and Rational Expressions

3

Lesson

1.1.1

In this Topic you’ll find out about the necessary conditions for rational numbers to be defined.

Topic

8.1.1 Fractions and Rational ExpressionsFractions and Rational Expressions

4

Lesson

1.1.1

Rational Expressions Can Be Written as Fractions

Topic

8.1.1

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.

34

31

89

1x + 1 2x + 1

x – 1

Examples of rational expressions are:

, , , , .


Rational expressions are written in the form , where q ≠ 0. pq

5

Lesson

1.1.1

An Expression is Undefined if the Denominator is Zero

Topic

8.1.1

If the denominator is equal to zero, then the expression is said to be undefined (see Topic 1.3.4).

2x + 1

x – 1So, 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 = –1 + 1 = 0

… and the expression would be undefined.


6

Example 1

Topic

8.1.1

Solution

It is undefined when the denominator x + 2 equals zero.

Solution follows…

This means that the expression is undefined when x = –2.

Determine the value of x for which

the expression is undefined.x + 2

7


7

Example 2

Topic

8.1.1

Solution

It’s undefined when the denominator x2 – 4 equals zero.

Solution follows…

So, solve x2 – 4 = 0 to find the values of x:

Determine the value(s) of x for which the

expression is undefined.x2 – 4

2

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


8

Example 3

Topic

8.1.1

Solution

Solution follows…

If the denominator equals zero, the expression is undefined.

Determine the value(s) of x for which the

expression is undefined.x2 – 8x + 15

7x

So, the expression is undefined when x = 3 or x = 5.

Factor the denominator to give: (x – 3)(x – 5)

7x

This happens when either (x – 3) or (x – 5) equals zero.


9

Determine the value(s) of the variables that make the following rational expressions undefined.

1. 2. 3.

4. 5. 6.

7. 8. 9.

Lesson

1.1.1

Guided Practice

Topic

8.1.1

x = –2

Solution follows…

4y – 13

8 + 4x3 – 2x

5x – 30x + 1

2x + 1x2 – 3x – 28

a2 – 2a + 1a2 + 7a + 12

24y2 + 11y – 3

– 4k + 526 + 11k – k2

x3 – 9xx2 + 1 2a3 – a2

3a3 – 6a2 – 45a

y = 1

4x = 6

a = –4, –3 y = , –31

4k = 13, –2

x = 0, 3, or –3 x = 7, –4 a = 0, 5, –3


10

Determine the value(s) of the variables that make the following rational expressions undefined.

1. 2. 3.

4. 5. 6.

7. Jane states that the rational expression is defined when x is any real number. Show that Jane is incorrect.

Independent Practice

Solution follows…

Topic

8.1.1

x = 2

2k – 13k3 – 3k2 – 18k

x – 4 – x2

3x – 6

3x + 13x2 – 2x – 1

y3 + y3y2 + 9y + 6

m + 5m2 + 8m +7

5x6x – x2 – x3

6x2 + 13x + 58x3 + 32x2 + 30x

y = –1, –2 m = –1, –7

k = 0, 3, or –2x = 0, 2, or –3 3

2x = 0, – , or – 5

2

x = – or x = 1 will make the denominator 0, so Jane is incorrect1

3


11

Topic

8.1.1

Round UpRound Up


This Topic about the limitations on rational numbers will help you when you’re dealing with fractions in later Topics.

In Topic 8.1.2 you’ll simplify rational expressions to their lowest terms.

1 topic 8.1.1 fractions and rational expressions fractions and rational expressions

Documents