Multiplying Rational Expressions

Multiplying Rational Expressions

What is a rational expression?

A rational expressions is a quotient of two polynomials.

 $f\left(x\right)\ =\ \frac{p\left(x\right)}{q\left(x\right)}$  

Multiple Choice

What is a rational expression?

• A quotient of two polynomials.

• The difference between two polynomilas.

• The sum of two polynomials.

• The product of two polynomials.

Operations with Polynomials

• We can add them.

• We can subtract them.

• We can multiply them.

• We can divide them.

• And any other mathematical iteration...

Multiple Select

What mathematical operations can be done to rational expressions?

• All mathematical operations.

• Some mathematical operations.

Multiplying Rational Expressions

• Step 1 See if any of the expressions can be simplified.

• Step 2 Multiply the numerators.

• Step 3 Multiply the denominators.

• Step 4 Simplify if you can.

Poll

I can simplify the following!

x^2 +5x -14

• Yes!

• No, no ma'am!

• Get me started!

Let's factor  $x^2+5x-14$  

Ask the questions:

1. What two numbers can I multiply together to get -14?
2. Of the numbers I've found, which ones when combined will give me a +5 (the coefficient of +5x)?

Answer:  $\left(x+7\right)\left(x-2\right)$  

Multiple Choice

How do the numbers -14 and +5 relate?

• -14 is a result of the numbers we multiplied and then combined to get +5

• -14 is a result of the numbers we combined and then multiplied to get +5

Let's remind ourselves of how to multiply fractions! After all Rational Expressions are the grown up version of fractions!

•  $\frac{1}{2}\cdot\frac{2}{5}=\frac{2}{10}\ let's\ reduce!\ =\ \frac{1}{5}$  

• Be mindful, we could have cancelled the 2's out first and then multiplied!

• Notice that we multiply STRAIGHT ACROSS!

Fill in the Blank

When we multiply fractions we multiply _____________ across.

Let's apply this to rational expressions!

Step 1. Multiply straight across, numerator with numerator, denominator with denominator!
Step 2. Distribute the x to the numerator.
Step 3. Apply FOIL to the denominator.
Step 4 Simplify if you can!

Example  $\frac{x+2}{x+3}\cdot\frac{x}{x+4}=\frac{\left(x+2\right)\left(x\right)}{\left(x+3\right)\left(x+4\right)}=\frac{\ x^2+2x}{x^2+7x+12}$  

Multiple Choice

When we multiply rational expressions we multiply...

• Straight across

• Diagonally

Let's do another one!

•  $\frac{x^2-9}{x^2-x-6}\cdot\frac{x-3}{x+3}=\frac{\left(x-3\right)\left(x+3\right)}{\left(x+2\right)\left(x-3\right)}\cdot\frac{\left(x-3\right)}{\left(x+3\right)}=\frac{x-3}{x+2}$  

•                Part 1                                      Part 2                      Part 3

• Part 1 this is the original problem, we see we need to factor.

• Part 2 we factored both quadratics!

• Part 3 cancel out like terms DIAGONALLY or numerator to denominator

Fill in the Blank

We _________ both the numerator and denominators first.

Multiple Choice

In order to get to our final answer, we

• Cancelled out like terms

• Cancelled out different terms

Open Ended

Which part of this lesson is the clearest for you to understand?

Poll

The part of the lesson where I learned the most was...

• FOIL

• Distribution

• Multiplying fractions

• Multiplying the rational expressions

• Simplifying the rational expressions