Lesson 4

Add, Subtract, multiply, and divide rational expressions

Warm up

Find the product.

• (x+6)(x-6)

• (7x-8)(7x+8)

1. (x+6)²

Warm up

• Divide 5x³ + 20x² - 10x by 5x

What is a rational expression?

• A rational expression is an expression that can be written as a ratio of two polynomials.

Ex.

8

-2x

x

x2 – 3x

x + 5

x² + 2x - 15

Excluded values

• A rational expression is undefined when the denominator is 0. A number that makes the denominator 0 is an excluded value.

• Let’s look at the examples before…are there any values that should be excluded?

8

-2x

x

x2 –3x

x + 2

x² + 2x - 15

Examples

• Find the excluded values, if any, of the expression.

1. 9x

5x - 15

2. x -1

x² - 16

3. x + 6

x² + 4x - 12

To find the excluded values, you have to factor the denominator (if possible). Then set each factor equal to ZERO and solve.

Simplest Form

• A rational expression is in simplest form if the numerator and denominator have no factors in common other than 1.

• Examples: Are these expressions in simplest form?

8

-2x

x

3x - 9

x + 5

x² + 2x - 15

What about these?

• Simplify, if possible. State the excluded values.

2m

8m(m-1)

11

y + 6

7q² - 14 q

14q²

Try these…

• Simplify and state the excluded values.

x² + 4x – 21

x² - 5x + 6

x² + 13x + 42

x² -2x -63

Guided Practice

• Pp. 437-438 1-3 (all parts)

• Rational Expressions Handout

Multiplying Rational Expressions

• ½ * ¾ =

• 2/5 * 5/4 =

• 2/3 * 5/4 =

• 2x² * 1/x =

• 4xy/5 * 20y/4x²

Multiply and Divide Rational Expressions with variables

• Factor the numerators and denominators.• Simplify where possible (…Cancel)• Multiply the numerators and denominators straight

across.• List restrictions (excluded values) if any…

Example: x² + x – 6 5x² + 15x

10x² - 20x x² - 2x - 15*

Examples

4x² (x+8)

2x³ + 10x – 48x*

x² - 1 4x – 2

2x² - 3x + 1 3x + 18 *

Guided Practice

• Textbook p. 441-442

Let’s Divide

• ¾ divided by ½

• 2/5 divided by 2

• 1/3 divided by 4/5

So…to divide rational expressions…

1. Find the reciprocal of the second fraction.

2. Simplify according to your rules…3. Multiply straight across…

Try these…

24 ÷ 6

5x² 25x²

7x +21 21x + 63

30 20÷

x + 2 x² + 11x + 18

3x – 3 x - 1

Graphic Organizer

• Fill in your graphic organizer “How do you multiply or divide rational expressions?”

• Work each example in the space provided.

Practice

• Complete textbook pp. 441-444

• Dividing rational expressions handout.

• Multiplying rational exp handout.

Finding the LCM

• Find the LCM of 6x and 8x²

Step 1: Write the factors of each expression.

6x = 2 * 3 * x

8x² = 2 * 2 * 2 * x * x

Next…

• Step 2:

6x = 2 * 3 * x

8x² = 2 * 2 *2 * x * x

• Step 3: List the common factors once…and then list every other factor…

• Step 4: MULTIPLY

2 * x * 3 * 2 * 2 * x =24 x²

Circle the factors that both

expressions have in common…

Guided Practice

• Least Common Multiple Practice Handout

• Complete 1-3

Find the LCM of…

• x² + 2x – 8 and x² + 7x + 12Step 1: List the factors of each polynomialx² + 2x – 8 = (x-2)(x+4)x² + 7x + 12 = (x+3)(x+4)Step 2: Circle the common factors.Step 3:List the common factor once and then

the other factors…then multiply…(x+4)(x-2)(x+3) There’s no need to multiply this out…Leave it as it is…

Guided Practice

• Complete the remaining problems on the practice handout.

Add/Subtract Rationals

• x + 3 + x – 2

7x 7x

• 5x + 7 - 2x – 9

3x – 4 3x – 4

So, on these I need to

add/subtract the numerators and just bring

over the denominator,

then simplify, if possible?

What if the denominators are not the same?

• 11 + 15 12x² 16x5

1. Find the LCD of the denominators…aka LCM.

2. Multiply top and bottom by missing factors…

3. Now that the denominators are the same…follow your steps for adding fractions…

Try these…

• 7 + 12 18r² 9r³

• 11 + 4 2x 7x

• 8 - 5 3x³ 12x

What to do with these…

• 12 - 4xx+2 x-3

• 4 - 3 x² - 7x x

• 2x + x+ 4x² - 3x x – 3

One more…

• 1 - 1

x² + 5x + 4 x² - 16

Fill in the graphic organizer for add/subt rational expressions.

Practice

• Pp. 439-440 Student Text

• More practice handouts 11.5 & 11.6