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 Blanks

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 Blanks

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