6-1 Simplifying Fractions

Simplest form for an algebraic fraction is when the numerator and denominator have no common factors other than 1 or –1.

D: How would you simplify ?

To simplify fractions, first factor the num. and den.

Then, look for common factors that equal 1 or –1

=

Finally, “restrict variables” so that the denominator cannot be zero. Restrictions are based on the original fraction, not just the simplified fraction.

Restrictions:

Example: Simplify

are there common factors?

Restrictions?

Go back to original fraction

Ø You try: Simplify

;

Ø Solve for x:

Factor both sides:

Divide by :


6-2/6-3 Multiplying and Dividing Fractions

Multiplying and dividing algebraic fractions is completed using the same rules as numeric fractions.

To multiply two fractions:

To divide two fractions you multiply the first by the reciprocal of the second. (recall the definition of division).

§  You can simplify before you multiply or after. Usually before you multiply saves you some steps.

Example: Multiply:

=

Example: Divide:

=

The rule of Exponents for Power of a Quotient states that:

How could you use this rule to simplify : ?

=

6-4 Least Common Denominator (LCD )

In order to add or subtract fractions, we need to be able to find the LCD.

D: How do we find the LCD of numeric fractions?

Find the Least Common Multiple of the denominators using prime factorization.

Find the LCD of algebraic fractions the same way:

Example: Find the LCD of

Note: The original fractions may or may not be simplified

Step 1: Factor both denominators:

Step 2: Find the product of the greatest powers of each factor.

D: How do you rewrite the fractions using the LCD?

It is often helpful to use the factored denominators:

What do we multiply the original denominator by to get the new one?

3, therefore we need to multiply the numerator by 3:

Ø You try:

Ø Rewrite using the LCD:

so the LCD is

6-5/6-6 Adding and Subtracting Fractions

Just like numeric fractions, when you add or subtract algebraic fractions you must first rewrite the fractions using a common denominator and then add or subtract numerators

Example: Subtract

Be very careful with subtraction: =

Example: Add Reminder

= Is it simplified?

Yes it is, but always ask that question.

Ø You try: What is

Ø What is

LCD =

=

Ø What is ?

6-7 Polynomial Long Division

D: How do we perform long division? What happens when we have a remainder? What did you learn to do with it in 6th grade math?

This also happens to be how you change an improper fraction to a mixed number.

Polynomial Long Division is similar. The focus of the division is on the variable term in the divisor and dividend.

Example (on board):

Example (on board):

§  Procedure for long division:

  1. Rewrite the dividend in order of decreasing degree of the variable.
  2. Add zero coefficient to the dividend for any missing terms.
  3. Ask “how many times does the variable portion of the divisor go into the first term of the dividend, or what is left after a subtraction.
  4. Multiply the term from step 3 by the divisor and subtract (make sure to line up like terms).
  5. Repeat 3 and 4 until there are no more variable terms
  6. Show the as an addition or subtraction from the quotient.

Example: